1 INTRODUCTION

This HTML document contains all the results and plots supporting the paper network structure shapes languages: disentangling the factors driving variation in communicative agents. All the data and scripts needed to reproduce this document are available in the GitHub repository.

1.1 Typographic conventions

This HTML document uses the following font and color conventions:

variables’ name are represented using italic text
emphasis is represented using bold text
software or programming concepts (e.g., applications, packages or function names) are represented using fixed font text
hyperlinks to sections of this document and to external resources on the web are represented as link to the Introduction or link to R project’s website

1.2 Software and hardware info

The full information about the version of R (R Core Team, 2022). The packages used to obtain this html file can be seen when opening the corresponding Rmarkdown file.

To perform the simulations, we used two different machines:

Latitude 7400, DEll, with a processor Intel64 Family 6 Model 142 Stepping 12 GenuineIntel, x64bits, 16.0 Go RAM. R was used with version 4.2.1 and 4.3.0
PO145, with a processor Intel(R) Core(TM) i5-8365U CPU @ 1.60GHz 1.90 GHz, x64bits, and 16.0 Go RAM. R was used with version 4.2.1.

2 DATA

For more information on the model, refer to the main paper.

2.1 The procedure

We conducted three types of simulations:

DATASET 1. In the first, we systematically compared two sets of networks differing in exactly one metric.
DATASET 2. In the second, we used other datasets (with automatically generated random, scale-free, and small-world networks using a wide range of parameters) to complement these results and look at:
- the correlations between the different metrics
- the relative contribution of the metrics by applying different types of models (machine learning models, such as random forests and support vector machines, as well as linear regression) using the different metrics normalized as predictors. These models help understand the relative contribution of each of these metrics in several network types. We expect to reach similar conclusions between network types and between methods.
DATASET 3. In the third, we looked at the language value and variation in heterogeneous populations (where some agents are locally biased) in all types of networks.

2.2 Step 1: generating datasets using Netlogo program

The Netlogo program is available in the GitHub repository associated to this Supplementary Materials (see my github under the name Netlogo_program_dirichlet.nlogo. This is an extension of the Netlogo program available in our previous paper “Interindividual variation refuses to go away: a Bayesian computer model of language change in communicative networks” (2021) published in Frontiers in Psychology, and available in the Supplementary Materials of this paper here. While the model is the same, a few functions have been added (for example, it is now possible to study multinomial language features).

To understand how to use our Netlogo code and parameters, please refer to the Netlogo guide, a pdf file that can be found in the same github folder (see appendix_netlogo.pdf file)

Using this Netlogo file, we generated networks with specific properties. The results of the simulations are stored in a Raw folder. They have the following name: DN_typesimulation_networktype:

N can take the values 1, 2, or 3 depending on the dataset number
typesimulation can take the values continuous or multinomial depending on the language studied
networktype can take the values scalefree, smallworld, or random based on the network used to generate the simulations.

For example, a file D1_dirichlet_random.csv present in the folder Raw is the raw file obtained from Netlogo (using the code called D1_dir_random in Behavioral Space), and includes data obtained using random networks, a multinomial language, for the dataset 1 (comparing sets of networks).

2.2.1 Parameters

Below are the specific parameters used to generate networks on Netlogo.

In Dataset 1, we conducted 100 replications for each combination of parameters. In Dataset 2, we generated 900 networks for each network type. This entailed 10 replications per combination for random networks, 5 replications per combination for small-world networks, and 100 replications per combination for scale-free networks. This choice was guided by the aim to maintain models with a similar amount of data. However, it is important to note that we also tried to study models with 100 replications for each combination of parameters within each network type (so much more data for small-world compared to scale-free networks), but this change did not impact any of our conclusions.

For scale-free networks:

Dataset number	Size network	Init value Dirichlet
Dataset 1	50; 100	0; 5; 20
Dataset 2	50; 150; 300	0; 5; 20
Dataset 3	150	0; 5; 20

For random networks:

Dataset number	Size network	Connection probability	Init value Dirichlet
Dataset 1	50; 100	0.06; 0.24; 0.2;5 0.26	0; 5; 20
Dataset 2	50; 150; 300	0.05; 0.15; 0.25; 0.35; 0.45; 0.55; 0.65; 0.75; 0.85; 0.95	0; 5; 20
Dataset 3	150	0.08	0; 5; 20

For small-world networks:

Dataset number	Size network	Number neighbors	Rewiring probability	Init value Dirichlet
Dataset 1	150	24	0.1; 0.9	0; 5; 20
Dataset 2	50 150	2; 4; 8; 24	0.1; 0.3; 0.5; 0.7; 0.9	0 5 20
Dataset 3	150	2	0.1; 0.9	0; 5; 20

2.2.2 Note on Initial language exposure terminology

The value for initial language exposure (which we refer here as init_lang_exp) tells the number of utterance that have been heard after the agent is born. Since the agents are born with a vector = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], then their value in $u_4$ becomes 6 (5+1) or 21 (20+1).

In the datasets, this variable has three modalities:

init_lang_exp = 0 (so $u_4$ at first = 1, similar to other values of the Dirichlet parameters): we call this condition without initial language exposure. In the paper, we refer to this situation as language emergence scenario.
init_lang_exp = 5 (so $u_4$ at first = 6, contrary to other Dirichlet values which = 1): we call this condition with (weak) initial language exposure. In the paper, we refer to this situation as language change scenario.
init_lang_exp = 20 (so $u_4$ at first = 21, contrary to other Dirichlet values which = 1): we call this condition with (strong) initial language exposure. In the paper, we do not refer to this condition, but that it is also a scenario of language change.

In these Supplementary Materials, we preferentially refer to this variable with the modalities “Without”, “With (weak)”, and “With (strong)” (instead of language change and emergence) because it allows us to contrast between the two language change conditions (weak: $u_4$=6 and strong: $u_4$=21).

2.3 Step 2: cleaning netlogo output files

After we performed the analysis in Netlogo, we obtained files which often contain thousands of simulations. Thus, these files are very heavy and hard to read, containing many unecessary informations (useless column, etc). (For example, for each network, we stored the final value of the Dirichlet vector for all agents!). We used another Rmarkdown file to clean this data (see CleanData.RmD in the GitHub folder). This file first selects only the columns containing meaningful information. Then, it computes the following:

2.3.1 DATASET 1

first, it computes the exponent of the scale-free distribution given its degree distribution (thus, this step was only performed in scale-free networks). This exponent is what we will call the variable “degree distribution” (important note: we called it so for simplicity reasons, but this variable does not actually refer to the degree distribution! it shows the exponent, which is an indicator of the shape of the degree distribution). All values are usually included between 2 and 3 as the scale-free networks were generated using Barabasi-Albert algorithm. However, networks with an exponent of 2 have a distribution less skewed compared to network with an exponent of 3.
second, it shows the density plots for the intersection of our different metrics (pathlength, assortativity, size, clustering coefficient, degree distribution): for example, we plot the values of the different simulations of pathlength in the x axis and the value of assortativity in the y axis. Thanks to these plots, we could visualize what values of the different metrics to select while keeping one constant. Then, we selected sets of 100 networks with the desired values (see the method in the main paper for more information). Please note that we used several types of networks to investigate the effect of all metrics. Indeed, it is not that easy to find networks where only one metric differs from the others! Thus, we used scale-free network to investigate the effects of pathlength, assortativity, size, and degree distribution, random network to disentangle the effects of mean degree and pathlength, and small-world network to investigate the effect of clustering coefficient.

For example: for a network with a size of 50 people and an initial language exposure = with (weak), here is what looks like the plot with pathlength in x and assortiativty in y (one point is the average value for one network). Example of the values of pathlength and assortativity for all the networks generated. The networks selected were inside the red circles.

To determine the thickness between the two black lines and the diameter of the red circles, we manually adjusted the values until we identified the smallest possible differences between our sets. It’s important to note that other values could also be suitable. We experimented with various alternatives, such as smaller values for the control variables, resulting in smaller differences in the variables of interest. However, these adjustments did not alter the main conclusions.

third, it computes the mean of each metric for each condition. This data is gathered in a table, which is then exported under a file ending by _tab_merge.csv.
fourth, we checked the maximum language value for each participant, for all replication. We save this dataframe in files ending by _lang_merge.csv.
fifth, we computed the measure of inter-individual variability (by estimating the Kullback-Leibler divergence on all pairs of agents) and intra-individual variability (by estimating the entropy for all agents, and then applying the mean or the standard deviation). We save this dataframe in file ending by _merge.csv. Please note that the computation of Kullback-Leibler divergence is quite long.

All these datasets were exported in a folder called Multinomial for simulations that used a multinomial language, and in a Continuous folder for simulations that used a continuous language.

2.3.2 DATASET 2 and DATASET 3

first, it computes the exponent of the scale-free distribution given its degree distribution.
second, we computed the measure of inter-individual variability (by estimating the Kullback-Leibler divergence on all pairs of agents) and intra-individual variability (by estimating the entropy for all agents, and then applying the mean or the standard deviation). We save this dataframe in files ending by _merge.csv (note that the computation of Kullback-Leibler divergence is quite long).

In these datasets, we did not manually selected sets of networks.

2.3.3 Summary of the results

Then, we aggregated these datasets into the same files. These files are the one used in this analysis and are available in the Summary folder:

Input file	Dataset	Type simulations	Type network	Information type
`TIMELINE_continuous_scalefree.csv`	Timeline	continuous language	scale-free	show the variation and language at each round in a few populations
`TIMELINE_dirichlet_scalefree.csv`	Timeline	dirichlet language	scale-free	show the variation and language at each round in a few populations
`DATASET1_continuous_table.csv`	Dataset 1	continuous language	scale-free, small-world, random	gather information about the metrics values for each set in a table
`DATASET1_continuous_variation.csv`	Dataset 1	continuous language	scale-free, small-world, random	gather information about the language and variation (inter and intra) for each simulation
`DATASET1_dirichlet_table.csv`	Dataset 1	dirichlet language	scale-free, small-world, random	gather information about the metrics values for each set in a table
`DATASET1_dirichlet_variation.csv`	Dataset 1	dirichlet language	scale-free, small-world, random	gather information about the variation (inter and intra) for each simulation
`DATASET1_dirichlet_language.csv`	Dataset 1	dirichlet language	scale-free, small-world, random	gather information about the language for each agent in each simulation
`DATASET2_dirichlet.csv`	Dataset 2	dirichlet language	scale-free, small-world (with different rewire probability and average number of neighbors), random (with different connection probability)	gather information about the variation (inter and intra) for each simulation
`DATASET3_dirichlet_variation.csv`	Dataset 3	dirichlet language	scale-free, small-world (with two differnt probability of rewire), random	gather information about the variation (inter and intra) for each simulation
`DATASET3_dirichlet_language.csv`	Dataset 3	dirichlet language	scale-free, small-world (with two differnt probability of rewire), random	gather information about the language for each simulation
`DATASET3_dirichlet_language_2.csv`	Dataset 3	dirichlet language	scale-free, small-world (with two differnt probability of rewire), random	gather information about the language (different type of language data) for each simulation
`DATASET3_supplementary.csv`	Dataset 3	dirichlet language	scale-free, small-world (with two differnt probability of rewire), random	structural information about the local metrics when the network is first build (no evolution of language with time)

3 RESULTS

Before looking at the results from the three datasets, we perform a preliminary analysis aiming at understanding when we should stop the simulations.

3.1 Timeline

As a reminder, here, time is discretized into iterations, starting with iteration 0 (the initial condition of the simulation) in increments of 1. At each new iteration, i > 0, all agents produce one utterance using their own internal representation of language and production mechanism (as described in the method section of the main paper). Thus, the language of the population evolves at each iterations. We first aim to look how to the language evolves over the course of iterations. This allows us to choose after which iterations we stop the simulations.

The stabilization time was already explored in our previous paper (Josserand, Allassonnière-Tang, Pellegrino, & Dediu, 2021), thus we did not go into detail in this process here. Stabilization time captures how long (in terms of interaction cycles) it takes for the language of a given network to reach a stable state. In our previous paper, we estimated the stabilization time based on the method developed in (Jannsen (2018)) (p. 79). To do so, we used a fixed-size sliding window within which we estimate the change in the language value, we multiply this number by 10,000, round it, and stop if this number is equal to zero (i.e., the slope is within ±0.00001 of 0.0) for 50 consecutive steps. Practically speaking, the maximum number of ticks of our model is nIterations = 5000, and the size of the sliding window is $W = nIterations/10$. For a given window, we estimated the change, $t(e_g)$ using the following formula, where $g$ is the number of iterations. \[t(e_g)=\frac{(e_{g+w}-e_g)}{W}*10000\] On the rounded $t(e_g)$ values, we find the first value of $g$, $g_{stabilization}$, when the rounded value of $t(e_g) = 0$, and we stop if for 50 consecutive steps (i.e., $g \in [g_{stabilization}..(g_{stabilization}+50)]$), there is no change, $t(e_g) = 0$ in this case, the stabilization time is the first moment where there was no change, namely $g_{stabilization}$.

In our previous paper, Figure 13 shows the stabilization time for different network types (random, scale-free, and small-world). In this paper, we observed the stabilization time for two groups of agents (biased and unbiased). Overall, we observed that the stabilization time was never higher than 1000 iterations for any of the networks or any of the groups.

Since the model used in this paper is very similar (except that the language used is not binary anymore but multinomial or continuous), we expect similar results. However, due to the change in the type of language, we ran a few extra simulations to make sure these conclusions are transferable.

3.1.1 Multinomial

We observe what happens with time (from round 1 to round 4000) in a scale-free network containing 150 nodes. We chose to look specifically at scale-free networks because it was the type of network that requires the most time to stabilize (see our previous paper -Josserand et al. (2021)). We observe the evolution of different measures: inter-individual variation, the mean intra-individual variation and the std of intra-individual variation.

We look here at 10 different simulations :

5 simulations in networks with 50 agents
5 simulations in networks with 150 agents

In the following graph, each simulation is characterized by its color. The vertical plain lines shows the stabilization value based on the method presented above. The vertical black dashed line shows our choice for selecting the final value of the language.

Figure 1. Looking at the evolution of our different measures (top: inter-individual variability middle: mean of intra-individual variability bottom: std of intra-individual variability) with time. All agents speaks once in one interaction. We observe here 10 different simulations. Each simulation has a specific color. The plain vertical line shows the stabilization time for each replication (corresponding colors). The dashed black line shows our choice for selecting the final value of the language.

According to this data, our measures of variation have always stabilized after 1500 interations.

However, we decided here to study the language after agents have interacted during 3000 iterations. Indeed, it does not require much more time to compute, and we make sure that the language will indeed be stabilized in all our simulations.

3.1.2 Continuous

We did exactly the same but using the language simulations where language is repsented as a continuous feature..

We observe the evolution of different measures: inter-individual variation, the mean intra-individual variation and the std of intra-individual variation.

We look here at 10 different simulations :

5 simulations in networks with 50 agents
5 simulations in networks with 150 agents

In the following graph, each simulation is characterized by its color.

Please note that in the first graph below (inter-individual variation), it seems that there is only one simulation (one color), but this is only the case because all simulations have almost the same values, so the points are superimposed.

Figure 2. Looking at the evolution of our different measures (top: inter-individual variability middle: mean of intra-individual variability bottom: std of intra-individual variability) with time. All agents speaks once in one interaction. We observe here 10 different simulations. Each simulation has a specific color. The plain vertical line shows the stabilization time for each replication (corresponding colors). The dashed black line shows our choice for selecting the final value of the language.

Popualtions with continuous language take way less time to stabilize. For simplicity reasons, we use the same threshold as for multinomial language.

Summary:

We record the final value of the language after 3000 iterations.

3.2 DATASET 1: comparing sets of networks

3.2.1 Multinomial

This part first presents the detailed results for each metric. However, you can refer to the Summary multinomial part, where you can see all the information in one plot.

3.2.1.1 Pathlength

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low pathlength set	3.374239	1.96	0	-0.3125999	2.798979	50	Scalefree
High pathlength set	4.987007	1.96	0	-0.3057365	2.790429	50	Scalefree

This should be interpreted in the following way:

there is a difference of 1.6128 between the two sets concerning pathlength
there are differences of 0 for the neighbors, 0 for the clustering coefficient, 0.0069 for the assortativity, -0.0085 for the exponent, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in pathlength, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 3. Looking at the variability inter- and intra-individual in sets varying in pathlength. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

We apply Cohen’s d test and a classic t.test, in networks without language exposure, with (weak) language exposure, and with (strong) language exposure. We gathered the results in a summary table:

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
pathlength	Inter	Without	1.464	4	1.150	1.778
pathlength	Inter	With (weak)	1.522	4	1.205	1.838
pathlength	Inter	With (strong)	1.247	4	0.942	1.552
pathlength	Mean Intra	Without	0.070	1	-0.209	0.349
pathlength	Mean Intra	With (weak)	0.308	2	0.588	0.027
pathlength	Mean Intra	With (strong)	0.023	1	-0.256	0.302
pathlength	Std Intra	Without	0.296	2	0.016	0.577
pathlength	Std Intra	With (weak)	0.677	3	0.390	0.964
pathlength	Std Intra	With (strong)	0.505	3	0.221	0.788

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
pathlength	Inter	Without	10.351	185.325	0.0000000	1.150	1.778
pathlength	Inter	With (weak)	10.760	173.081	0.0000000	1.205	1.838
pathlength	Inter	With (strong)	8.819	194.129	0.0000000	0.942	1.552
pathlength	Mean Intra	Without	0.493	183.826	0.6226403	-0.209	0.349
pathlength	Mean Intra	With (weak)	-2.176	183.850	0.0308327	-0.588	-0.027
pathlength	Mean Intra	With (strong)	-0.163	177.904	0.8706430	-0.302	0.256
pathlength	Std Intra	Without	2.095	194.168	0.0374853	0.016	0.577
pathlength	Std Intra	With (weak)	4.789	196.574	0.0000033	0.390	0.964
pathlength	Std Intra	With (strong)	3.568	192.189	0.0004535	0.221	0.788

3.2.1.2 Assortativity

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low assortativity set	4.007469	1.96	0	-0.4832008	2.747048	50	Scalefree
High assortativity set	4.016071	1.96	0	-0.1650383	2.752900	50	Scalefree

This should be interpreted in the following way:

there is a difference of 0.3182 between the two sets concerning assortativity
there are differences of 0 for the neighbors, 0 for the clustering coefficient, 0.0086 for the pathlength, 0.0059 for the exponent, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in assortativity, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 4. Looking at the variability inter- and intra-individual in sets varying in assortativity. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

We want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
assortativity	Inter	Without	0.405	2	0.686	0.123
assortativity	Inter	With (weak)	0.392	2	0.673	0.110
assortativity	Inter	With (strong)	0.140	1	-0.139	0.419
assortativity	Mean Intra	Without	0.467	2	0.184	0.750
assortativity	Mean Intra	With (weak)	0.083	1	-0.196	0.362
assortativity	Mean Intra	With (strong)	0.023	1	-0.256	0.302
assortativity	Std Intra	Without	0.234	2	-0.046	0.514
assortativity	Std Intra	With (weak)	0.050	1	-0.229	0.329
assortativity	Std Intra	With (strong)	0.034	1	-0.245	0.313

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
assortativity	Inter	Without	-2.861	187.026	0.0047091	-0.686	-0.123
assortativity	Inter	With (weak)	-2.771	192.766	0.0061313	-0.673	-0.110
assortativity	Inter	With (strong)	-0.988	181.943	0.3244111	-0.419	0.139
assortativity	Mean Intra	Without	3.303	174.885	0.0011619	0.184	0.750
assortativity	Mean Intra	With (weak)	0.587	196.558	0.5580783	-0.196	0.362
assortativity	Mean Intra	With (strong)	-0.165	194.361	0.8694981	-0.302	0.256
assortativity	Std Intra	Without	-1.655	192.451	0.0996226	-0.514	0.046
assortativity	Std Intra	With (weak)	0.355	188.545	0.7227121	-0.229	0.329
assortativity	Std Intra	With (strong)	0.242	175.069	0.8089962	-0.245	0.313

3.2.1.3 Exponent - Degree distribution

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low exponent set	4.101233	1.96	0	-0.3545311	2.552332	50	Scalefree
High exponent set	4.093472	1.96	0	-0.3459061	3.037600	50	Scalefree

This should be interpreted in the following way:

there is a difference of 0.4853 between the two sets concerning exponent
there are differences of 0 for the neighbors, 0 for the clustering coefficient, -0.0078 for the pathlength, 0.0086 for the assortativity, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in exponent, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 5. Looking at the variability inter- and intra-individual in sets varying in exponent. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
exponent	Inter	Without	0.313	2	0.593	0.032
exponent	Inter	With (weak)	0.330	2	0.611	0.049
exponent	Inter	With (strong)	0.152	1	-0.128	0.431
exponent	Mean Intra	Without	0.081	1	-0.198	0.360
exponent	Mean Intra	With (weak)	0.026	1	-0.252	0.305
exponent	Mean Intra	With (strong)	0.076	1	-0.203	0.355
exponent	Std Intra	Without	0.040	1	-0.239	0.319
exponent	Std Intra	With (weak)	0.316	2	0.597	0.035
exponent	Std Intra	With (strong)	0.120	1	-0.159	0.399

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
exponent	Inter	Without	-2.212	191.145	0.0281752	-0.593	-0.032
exponent	Inter	With (weak)	-2.332	195.703	0.0206957	-0.611	-0.049
exponent	Inter	With (strong)	-1.077	197.651	0.2827301	-0.432	0.127
exponent	Mean Intra	Without	0.571	197.918	0.5688533	-0.198	0.360
exponent	Mean Intra	With (weak)	-0.181	197.846	0.8569108	-0.304	0.253
exponent	Mean Intra	With (strong)	0.537	190.474	0.5920046	-0.203	0.355
exponent	Std Intra	Without	-0.285	196.407	0.7757957	-0.319	0.239
exponent	Std Intra	With (weak)	-2.235	196.847	0.0265320	-0.597	-0.035
exponent	Std Intra	With (strong)	0.846	197.731	0.3984539	-0.159	0.399

3.2.1.4 Size

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low size set	4.490076	1.96	0	-0.2974416	2.848549	50	Scalefree
High size set	4.499676	1.98	0	-0.2908039	2.851606	100	Scalefree

This should be interpreted in the following way:

there is a difference of 50 between the two sets concerning size
there are differences of 0.02 for the neighbors, 0 for the clustering coefficient, 0.0096 for the pathlength, 0.0066 for the assortativity, and 0.0031 for the exponent

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in size, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 6. Looking at the variability inter- and intra-individual in sets varying in size. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
size	Inter	Without	0.894	4	0.601	1.186
size	Inter	With (weak)	0.492	2	0.209	0.775
size	Inter	With (strong)	0.304	2	0.024	0.585
size	Mean Intra	Without	0.112	1	-0.167	0.391
size	Mean Intra	With (weak)	0.040	1	-0.238	0.319
size	Mean Intra	With (strong)	0.104	1	-0.175	0.383
size	Std Intra	Without	0.381	2	0.100	0.663
size	Std Intra	With (weak)	0.340	2	0.059	0.621
size	Std Intra	With (strong)	0.042	1	-0.237	0.321

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
size	Inter	Without	6.321	197.778	0.0000000	0.601	1.186
size	Inter	With (weak)	3.479	197.998	0.0006180	0.209	0.775
size	Inter	With (strong)	2.153	192.077	0.0325857	0.024	0.585
size	Mean Intra	Without	-0.792	195.438	0.4294977	-0.391	0.167
size	Mean Intra	With (weak)	0.286	188.377	0.7749995	-0.238	0.319
size	Mean Intra	With (strong)	0.734	190.333	0.4639971	-0.175	0.383
size	Std Intra	Without	2.695	188.560	0.0076647	0.100	0.663
size	Std Intra	With (weak)	2.406	192.068	0.0170732	0.059	0.621
size	Std Intra	With (strong)	0.298	195.277	0.7660905	-0.237	0.321

3.2.1.5 Clustering coefficient

These sets of networks were obtained using small-world networks, and varying the parameters rewire.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low clustering set	1.677852	48	0.3189198	-0.0145174	NA	150	Small-world
High clustering set	1.682844	48	0.5826920	-0.0065716	NA	150	Small-world

This should be interpreted in the following way:

there is a difference of NA between the two sets concerning clustering coefficient
there are differences of NA for the neighbors, NA for the pathlength coefficient, NA for the assortativity, NA for the exponent, and NA for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in clustering, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 7. Looking at the variability inter- and intra-individual in sets varying in clustering. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Please note that the x scale here is different due to differences in the network type.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
clustering	Inter	Without	4.163	4	3.667	4.660
clustering	Inter	With (weak)	4.160	4	3.664	4.656
clustering	Inter	With (strong)	3.420	4	2.983	3.858
clustering	Mean Intra	Without	0.050	1	-0.229	0.329
clustering	Mean Intra	With (weak)	0.087	1	-0.192	0.366
clustering	Mean Intra	With (strong)	0.296	2	0.576	0.015
clustering	Std Intra	Without	2.017	4	1.674	2.359
clustering	Std Intra	With (weak)	2.220	4	1.866	2.575
clustering	Std Intra	With (strong)	2.070	4	1.724	2.415

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
clustering	Inter	Without	29.440	99.004	0.0000000	3.667	4.660
clustering	Inter	With (weak)	29.416	99.007	0.0000000	3.664	4.656
clustering	Inter	With (strong)	24.184	99.015	0.0000000	2.983	3.858
clustering	Mean Intra	Without	0.356	193.893	0.7223852	-0.229	0.329
clustering	Mean Intra	With (weak)	0.618	197.975	0.5369757	-0.192	0.366
clustering	Mean Intra	With (strong)	-2.092	194.149	0.0377403	-0.576	-0.015
clustering	Std Intra	Without	14.259	99.958	0.0000000	1.674	2.359
clustering	Std Intra	With (weak)	15.701	99.653	0.0000000	1.866	2.575
clustering	Std Intra	With (strong)	14.635	99.254	0.0000000	1.724	2.415

3.2.1.6 Node degree

These sets of networks were obtained using random networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low node degree set	1.785524	11.97893	0.2450417	0.0138245	NA	50	Random
High node degree set	1.777197	12.69280	0.2531758	0.0249452	NA	50	Random

This should be interpreted in the following way:

there is a difference of -0.0068 between the two sets concerning node degree
there are differences of 0.0083 for the clustering coefficient, 0.3735 for the pathlength coefficient, 3^{-4} for the assortativity, NA for the exponent, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in node degree, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 8. Looking at the variability inter- and intra-individual in sets varying in node degree. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Please note that the x scale here is different due to differences in the network type.

Here, we want to understand whether the effect size under these differences is high or low.

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
node_degree	Inter	Without	0.708	3	0.995	0.420
node_degree	Inter	With (weak)	0.725	3	1.013	0.437
node_degree	Inter	With (strong)	0.648	3	0.935	0.362
node_degree	Mean Intra	Without	0.140	1	-0.139	0.420
node_degree	Mean Intra	With (weak)	0.268	2	-0.012	0.548
node_degree	Mean Intra	With (strong)	0.311	2	0.592	0.031
node_degree	Std Intra	Without	0.057	1	-0.222	0.336
node_degree	Std Intra	With (weak)	0.138	1	-0.141	0.418
node_degree	Std Intra	With (strong)	0.298	2	0.578	0.017

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
node_degree	Inter	Without	-5.006	191.822	0.0000013	-0.995	-0.420
node_degree	Inter	With (weak)	-5.125	197.972	0.0000007	-1.013	-0.437
node_degree	Inter	With (strong)	-4.585	195.159	0.0000081	-0.935	-0.362
node_degree	Mean Intra	Without	-0.987	197.253	0.3248058	-0.419	0.140
node_degree	Mean Intra	With (weak)	-1.894	191.699	0.0597234	-0.548	0.012
node_degree	Mean Intra	With (strong)	-2.202	197.439	0.0288564	-0.592	-0.031
node_degree	Std Intra	Without	0.406	196.701	0.6854013	-0.222	0.336
node_degree	Std Intra	With (weak)	-0.974	197.709	0.3313699	-0.417	0.142
node_degree	Std Intra	With (strong)	-2.105	191.655	0.0365533	-0.578	-0.017

3.2.1.7 Pathlength (using random)

These sets of networks were obtained using random networks. Pathlength was already explored using scale-free networks, and controlling for many metrics Here, we just want to observe the effect of pathlength in opposite to the effect of mean degree, in random network.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low pathlength set	4.314229	2.467467	0.0285249	-0.1041842	NA	50	Random
High pathlength set	4.314229	2.467467	0.0285249	-0.1041842	NA	50	Random

This should be interpreted in the following way:

there is a difference of 0.3735 between the two sets concerning pathlength
there are differences of 0.0083 for the clustering coefficient, -0.0068 for the node degree coefficient, 3^{-4} for the assortativity, and 0 for the size

We look at the differences between the two sets:

Figure 9. Looking at the variability inter- and intra-individual in sets varying in pathlength (using random networks, and not scale-free networks). The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Please note that the x scale here is different due to differences in the network type.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
pathlength_ran	Inter	Without	0.427	2	0.145	0.709
pathlength_ran	Inter	With (weak)	0.727	3	0.439	1.014
pathlength_ran	Inter	With (strong)	0.512	3	0.229	0.796
pathlength_ran	Mean Intra	Without	0.001	1	-0.278	0.279
pathlength_ran	Mean Intra	With (weak)	0.000	1	0.279	0.279
pathlength_ran	Mean Intra	With (strong)	0.023	1	-0.256	0.302
pathlength_ran	Std Intra	Without	0.134	1	-0.145	0.413
pathlength_ran	Std Intra	With (weak)	0.380	2	0.099	0.661
pathlength_ran	Std Intra	With (strong)	0.265	2	-0.015	0.545

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
pathlength_ran	Inter	Without	3.022	196.497	0.0028414	0.145	0.709
pathlength_ran	Inter	With (weak)	5.137	183.653	0.0000007	0.439	1.014
pathlength_ran	Inter	With (strong)	3.623	185.944	0.0003750	0.229	0.796
pathlength_ran	Mean Intra	Without	0.004	193.154	0.9969390	-0.278	0.279
pathlength_ran	Mean Intra	With (weak)	0.001	192.635	0.9990379	-0.279	0.279
pathlength_ran	Mean Intra	With (strong)	0.161	196.648	0.8721028	-0.256	0.302
pathlength_ran	Std Intra	Without	0.949	198.000	0.3438721	-0.145	0.413
pathlength_ran	Std Intra	With (weak)	2.686	196.273	0.0078432	0.099	0.661
pathlength_ran	Std Intra	With (strong)	1.872	196.351	0.0627478	-0.015	0.545

3.2.1.8 Summary multinomial

Plot with full data:

You can find below the plot used in the main paper, which shows exactly the same information except that we do not present the results for initial language exposure = With (Strong). Also, we renamed the initial language exposure with type of language scenario (without initial language exposure -> Language emergence and with (weak) initial language exposure -> Language change). See Note on Initial language exposure terminology for more information.

3.2.2 Continuous

This part shows exactly the same type of results, except that here language is continuous and not multinomial. We present the detailed results for each metric. However, you can refer to the Summary continuous part, where you can see all the information in one plot. Please also note that we do not contrast language emergence and language change here.

3.2.2.1 Pathlength

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low pathlength set	3.392992	2	0	-0.3101820	2.783516	50	Scalefree
High pathlength set	4.982433	2	0	-0.3068284	2.770500	50	Scalefree

This should be interpreted in the following way:

there is a difference of 1.5894 between the two sets concerning pathlength
there are differences of 0 for the neighbors, 0 for the clustering coefficient, 0.0034 for the assortativity, -0.013 for the exponent, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in pathlength, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 10. Looking at the variability inter- and intra-individual in sets varying in pathlength in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
pathlength	Inter	None	6.427	large	7.119	5.734
pathlength	Mean Intra	None	0.676	medium	0.389	0.963
pathlength	Std Intra	None	3.292	large	2.864	3.720

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
pathlength	Inter	None	-45.444	171.791	0.0e+00	7.119	5.734
pathlength	Mean Intra	None	4.781	187.838	3.5e-06	0.389	0.963
pathlength	Std Intra	None	23.279	189.202	0.0e+00	2.864	3.720

3.2.2.2 Assortativity

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low assortativity set	3.998033	2	0	-0.4584237	2.781426	50	Scalefree
High assortativity set	4.008808	2	0	-0.1753525	2.785640	50	Scalefree

This should be interpreted in the following way:

there is a difference of 0.2831 between the two sets concerning assortativity
there are differences of 0 for the neighbors, 0 for the clustering coefficient, 0.0108 for the pathlength, 0.0042 for the exponent, and 0 for the size

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in assortativity, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 11. Looking at the variability inter- and intra-individual in sets varying in assortativity in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

COhen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
assortativity	Inter	None	5.139	large	5.717	4.561
assortativity	Mean Intra	None	0.232	small	0.512	0.048
assortativity	Std Intra	None	2.174	large	1.822	2.526

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf
assortativity	Inter	None	-36.338	188.901	0.0000000	0.000
assortativity	Mean Intra	None	-1.643	190.737	0.1021278	0.003
assortativity	Std Intra	None	15.373	192.193	0.0000000	0.000

3.2.2.3 Exponent

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low exponent set	4.003612	2	0	-0.2662336	2.610069	50	Scalefree
High exponent set	3.988955	2	0	-0.2615309	3.067970	50	Scalefree

This should be interpreted in the following way:

there is a difference of 0.4579 between the two sets concerning exponent
there are differences of 0 for the neighbors, 0 for the clustering coefficient, -0.0147 for the pathlength, 0.0047 for the assortativity, and 0 for the size

We look at the differences between the two sets:

Figure 12. Looking at the variability inter- and intra-individual in sets varying in exponent in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
exponent	Inter	None	2.357	large	2.720	1.994
exponent	Mean Intra	None	0.033	negligible	0.312	0.246
exponent	Std Intra	None	3.359	large	3.792	2.926

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Inf	CI_Sup
exponent	Inter	None	-16.667	190.982	0.0000000	0.000	0.000
exponent	Mean Intra	None	-0.233	197.326	0.8157111	0.002	0.001
exponent	Std Intra	None	-23.753	176.871	0.0000000	0.000	0.000

3.2.2.4 Size

These sets of networks were obtained using scale-free networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low size set	4.428812	2	0	-0.2983585	2.845419	50	Scalefree
High size set	4.439875	2	0	-0.2902309	2.854948	100	Scalefree

This should be interpreted in the following way:

there is a difference of 50 between the two sets concerning size
there are differences of 0 for the neighbors, 0 for the clustering coefficient, 0.0111 for the pathlength, 0.0081 for the assortativity, and 0.0095 for the exponent

Thus, we compare the differences in inter and intra-individual variation between the two sets. We believe that these differences will reflect the differences in size, as the other metrics are kept (almost) constant.

We look at the differences between the two sets:

Figure 13. Looking at the variability inter- and intra-individual in sets varying in size in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
size	Inter	None	4.608	large	4.075	5.141
size	Mean Intra	None	0.213	small	0.067	0.493
size	Std Intra	None	2.871	large	3.269	2.474

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue	CI_Sup
size	Inter	None	32.586	177.690	0.0000000	0.000
size	Mean Intra	None	1.505	187.882	0.1339995	0.003
size	Std Intra	None	-20.304	189.611	0.0000000	0.000

3.2.2.5 Clustering coefficient

These sets of networks were obtained using small-world networks, and varying the parameters rewire.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low clustering set	1.677852	48	0.3191316	-0.0151775	NA	150	Small-world
High clustering set	1.682726	48	0.5820881	-0.0041973	NA	150	Small-world

This should be interpreted in the following way:

there is a difference of NA between the two sets concerning clustering coefficient
there are differences of NA for the neighbors, NA for the pathlength coefficient, NA for the assortativity, NA for the exponent, and NA for the size

We look at the differences between the two sets:

Figure 14. Looking at the variability inter- and intra-individual in sets varying in clustering coefficient in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Please note that the x scale here is different due to differences in the network type.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
clustering	Inter	None	9.538	large	10.519	8.557
clustering	Mean Intra	None	2.721	large	2.334	3.108
clustering	Std Intra	None	11.598	large	12.775	10.421

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	CI_Inf	CI_Sup
clustering	Inter	None	-67.443	109.604	0.000	0.000
clustering	Mean Intra	None	19.242	99.004	0.001	0.001
clustering	Std Intra	None	-82.009	141.684	0.000	0.000

3.2.2.6 Node degree

These sets of networks were obtained using random networks.

We first observe the values of the two sets:

	pathlength	neighbors	clustering	assortativity	exponent	size	network_type
Low node degree set	1.785351	12.0072	0.244033	0.0116981	NA	50	Random
High node degree set	1.785351	12.0072	0.244033	0.0116981	NA	50	Random

This should be interpreted in the following way:

there is a difference of -0.0032 between the two sets concerning node degree
there are differences of 0.0024 for the clustering coefficient, 0.1377 for the pathlength coefficient, 0.0021 for the assortativity, NA for the exponent, and 0 for the size

We look at the differences between the two sets:

Figure 15. Looking at the variability inter- and intra-individual in sets varying in node degree in network with a continuous language. The first graph shows the value of the language: it shows the favorite utterance for each agents in the three types of population (variable initial language exposure). The second graph shows the inter-individual variation (the x scale is the average Kullback-Leibler divergence on all pairs of agents). The third and fourth graph show respectively the mean and standard deviation on the intra-individual variation for all agents (the x scale shows respectively the mean and the std of the entropy on the Dirichlet internal representation of agents). In all the following graphs, green shows high value in the metrics while yellow shows low value in the metric.

Please note that the x scale here is different due to differences in the network type.

Here, we want to understand whether the effect size under these differences is high or low.

Cohen’s D:

Measure	TypeVariation	TypeLangage	CohenD	Magnitude	CI_Inf	CI_Sup
node_degree	Inter	None	2.183	large	2.535	1.830
node_degree	Mean Intra	None	0.038	negligible	0.240	0.317
node_degree	Std Intra	None	0.981	large	0.686	1.277

T-Test:

Measure	TypeVariation	TypeLangage	T_value	DF	PValue
node_degree	Inter	None	-15.433	159.110	0.0000000
node_degree	Mean Intra	None	0.272	196.039	0.7860134
node_degree	Std Intra	None	6.939	116.802	0.0000000

3.2.2.7 Summary continuous

3.3 DATASET 2: looking at correlations and models

In this dataset, we generated thousands of simulations using random, small-world and scalefree networks. Using this dataset, we aim to:

look at the correlations between our metrics
investigate the relative strength of each metric using models

While in dataset 1 we use specific sets of networks (which can be seen as “unnatural”), in Dataset 2, we observe “natural” networks (namely, we did not choose to exclude any of the networks generated). Please note that here we only analyzed a multinomial language (contrary to Dataset 1, we do not study data from networks with continuous language). You can refer to Parameters to know more about the parameters which were investigated, but please note that a summary (and data distribution) of the results is also presented below.

3.3.1 Data distribution

Please note that exponent is a metric considered only in scale-free networks. Additionnally, node degree and clustering coefficient are considered only in small-world and random network, since they do not vary in scale-free networks. We look at the data for each network type.

Random networks

size	clustering	node_degree	pathlength	assortativity	init_lang_exp	connection_prob	inter_var	mean_intra_var	std_intra_var
50 :300	Min. :0.0000	Min. : 2.20	Min. :1.043	Min. :-0.241523	With (strong):298	Min. :0.0500	Min. :6.300e-08	Min. :0.7455	Min. :3.542e-05
150:295	1st Qu.:0.2506	1st Qu.: 22.71	1st Qu.:1.250	1st Qu.:-0.031982	With (weak) :298	1st Qu.:0.2500	1st Qu.:1.264e-06	1st Qu.:1.3053	1st Qu.:5.088e-04
300:300	Median :0.5294	Median : 51.09	Median :1.469	Median :-0.013001	Without :299	Median :0.5500	Median :4.474e-06	Median :1.9657	Median :1.192e-03
NA	Mean :0.5010	Mean : 83.12	Mean :1.620	Mean :-0.021829	NA	Mean :0.5015	Mean :3.374e-04	Mean :1.8145	Mean :3.937e-03
NA	3rd Qu.:0.7495	3rd Qu.:126.51	3rd Qu.:1.751	3rd Qu.:-0.005664	NA	3rd Qu.:0.7500	3rd Qu.:1.702e-05	3rd Qu.:2.2386	3rd Qu.:2.611e-03
NA	Max. :0.9568	Max. :284.75	Max. :4.985	Max. : 0.096417	NA	Max. :0.9500	Max. :1.957e-02	Max. :2.3002	Max. :1.242e-01

Quick note: the number of random network is not exactly equal to 900 because we removed a few networks where there were isolated components.

In random networks, the only structural parameters we manipulated were the connection probability (from 0.05 to 0.95 in steps of 0.05) and the size (50, 150, and 300).

Figure 16. Distribution of the data in random networks.

Scale-free networks

size	exponent	pathlength	assortativity	init_lang_exp	inter_var	mean_intra_var	std_intra_var
50 :300	Min. :2.416	Min. :2.820	Min. :-0.5590	With (strong):300	Min. :0.002436	Min. :0.9714	Min. :0.01744
150:300	1st Qu.:2.887	1st Qu.:4.218	1st Qu.:-0.2991	With (weak) :300	1st Qu.:0.008562	1st Qu.:1.2778	1st Qu.:0.04448
300:300	Median :3.026	Median :4.980	Median :-0.2310	Without :300	Median :0.020193	Median :1.9350	Median :0.08100
NA	Mean :2.998	Mean :4.941	Mean :-0.2495	NA	Mean :0.020313	Mean :1.7960	Mean :0.07720
NA	3rd Qu.:3.122	3rd Qu.:5.645	3rd Qu.:-0.1879	NA	3rd Qu.:0.029502	3rd Qu.:2.2091	3rd Qu.:0.10310
NA	Max. :3.367	Max. :7.285	Max. :-0.0873	NA	Max. :0.047241	Max. :2.2538	Max. :0.17089

In scale-free networks, the only structural parameter we manipulated was the size (50, 150, and 300).

Figure 17. Distribution of the data in scale-free networks.

Small-world networks

size	clustering	node_degree	pathlength	assortativity	init_lang_exp	rewire	inter_var	mean_intra_var	std_intra_var
50 :300	Min. :0.001556	Min. : 4	Min. :1.020	Min. :-0.29315	With (strong):300	Min. :0.1	Min. :2.030e-07	Min. :0.8495	Min. :0.0001095
150:300	1st Qu.:0.100506	1st Qu.: 7	1st Qu.:1.829	1st Qu.:-0.06832	With (weak) :300	1st Qu.:0.3	1st Qu.:4.184e-05	1st Qu.:1.2909	1st Qu.:0.0028076
300:300	Median :0.232127	Median :12	Median :2.364	Median :-0.03556	Without :300	Median :0.5	Median :3.634e-04	Median :1.9660	Median :0.0087059
NA	Mean :0.296572	Mean :19	Mean :2.635	Mean :-0.04476	NA	Mean :0.5	Mean :2.291e-03	Mean :1.8188	Mean :0.0184484
NA	3rd Qu.:0.399529	3rd Qu.:24	3rd Qu.:3.071	3rd Qu.:-0.01115	NA	3rd Qu.:0.7	3rd Qu.:2.709e-03	3rd Qu.:2.2501	3rd Qu.:0.0275223
NA	Max. :0.980213	Max. :48	Max. :7.277	Max. : 0.11942	NA	Max. :0.9	Max. :3.078e-02	Max. :2.2993	Max. :0.1244774

In small-world networks, the only structural parameters we manipulated were the rewiring probability (0.1, 0.3, 0.5, 0.7, 0.9), the node degree (4, 8, 16, 48 neighbors), and the size (50, 150, and 300).

For simplicity reasons, we do not plot here the node degree as a facet.

$**Figure 18.** Distribution of the data in small-world networks.$

Figure 18. Distribution of the data in small-world networks.

3.3.2 Correlations

The data is not normally distributed. So we should use Spearman’s correlations instead of Pearson’s correlations. Since it does not cost much to compute both, you can find below both of them :)

Please note that I didn’t manage to put them in grid.arrange all together.

Pearson’s correlations:

Figure 19. Pearsons correlations.

Spearman’s correlations:

Figure 20. Spearmans correlations.

We found high correlations between some variables. In order to understand the direction of correlation, we graphically print the relationships between the metrics in all types of networks. Size will always be printed in a color layer.

Random networks:

Figure 21. Graphical visualization of the relationship between the metrics using a range of random networks.

Please note that we ran several simulation with random network using different connection probability and size of the network, which correspond to the different bumps in the visualization.

Small-world networks:

Figure 22. Graphical visualization of the relationship between the metrics using a range of small-world networks.

Please note that we ran several simulation with small-world network using different rewiring probability, node degree, and size of the network, which correspond to the different bumps in the visualization.

Scale-free networks:

Figure 23. Graphical visualization of the relationship between the metrics using a range of scale-free networks.

3.3.3 Principal Component Analysis

We also look at the PCA between these metrics in all types of networks.

Random networks:

Figure 24. Histogram of the eigenvalues of the PCA between the metrics using a range of random networks.

Figure 25. Contribution of the different metrics in the PCA using a range of random networks.

Small-world networks:

Figure 27. Histogram of the eigenvalues of the PCA between the metrics using a range of small-world networks.

Figure 28. Contribution of the different metrics in the PCA using a range of small-world networks.

Scale-free networks:

Figure 30. Histogram of the eigenvalues of the PCA between the metrics using a range of scale-free networks.

Figure 31. Contribution of the different metrics in the PCA using a range of scale-free networks.

3.3.4 Machine learning (RF, CF & SVM)

We predict variation (inter, mean intra, and std intra) from a collection of potential predictors using three machine learning techniques:

random forests (RF): as implemented by randomForest() in package randomForest. This function implements Breiman’s random forest algorithm (based on Breiman and Cutler’s original Fortran code); see here for more information. RF have been shown to be particularly relevant when predictors are inter-correlated (see Levshina, 2021, in the Chapter 25 “Conditional Inference Trees and Random Forests”). We extract the predictor importance using two indexes: the accuracy-based index and the gini-based index. The accuracy-based index measures the reduction in model accuracy if a particular predictor is removed or permuted. On the other hand, the Gini Index is a measure of impurity or purity in a decision tree. In the context of random forests, namely, it measures how often a randomly chosen element from the dataset would be incorrectly labeled if it was randomly labeled according to the distribution of labels in the node.
conditional random forests (CF): as implemented by cforest() in package partykit. The number of trees is 200, and the number of permutations is 10. We measured the predictor importance using the unconditional index. The unconditional index for a given predictor measures how well that predictor alone, in isolation, can help to improve the predictive accuracy of the model.
support vector machines (SVM): as implemented by fit(...,model="svm") in the rminer package; see here). We used a regression task (since the output is continuous). This function uses ksvm from kernlab package; more information on this function can be found here. We measured the predictor importance based on sensitivity analysis. The sensitivity method for predictor importance in SVM involves perturbing individual predictors to observe the resulting changes in the model’s output, quantifying the impact, and ranking predictors based on their sensitivity scores to identify the most influential features in the SVM decision-making process.

For each method, we extract some success measures, here the R² and RMSE using postResample function of rminer package. Note that we z-scored all the predictors.

3.3.4.1 Random networks

Success in language emergence context:

Inter-individual variation:

random forests: R² = 91.5% ±0.6%, RMSE = 0.001 ±0.000
conditional random forests: R² = 85.5% ±1.2%, RMSE = 0.001 ±0.000.
support vector machines: R² = 80.4% ±23.4%, RMSE = 0.001 ±0.001.

Mean Intra-individual variation:

random forests: R² = 45.3% ±0.5%, RMSE = 0.029 ±0.000
conditional random forests: R² = 59.1% ±0.2%, RMSE = 0.025 ±0.000.
support vector machines: R² = 46.8% ±8.3%, RMSE = 0.028 ±0.005.

Std Intra-individual variation:

random forests: R² = 76.3% ±0.8%, RMSE = 0.003 ±0.000
conditional random forests: R² = 79.0% ±0.6%, RMSE = 0.003 ±0.000.
support vector machines: R² = 83.1% ±11.7%, RMSE = 0.003 ±0.003.

Let’s plot this data:

Figure 33. Success (using R² and RMSE indexes) in random networks in language emergence scenarios for each type of method.

Predictor importance in language emergence context:

Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

Figure 34. Predictor importance in random networks in language emergence scenarios for each type of method.

Success in language change context:

Inter-individual variation:

random forests: R² = 95.9% ±0.4%, RMSE = 0.000 ±0.000
conditional random forests: R² = 82.9% ±1.2%, RMSE = 0.001 ±0.000.
support vector machines: R² = 88.5% ±15.0%, RMSE = 0.001 ±0.001.

Mean Intra-individual variation:

random forests: R² = 2.5% ±0.7%, RMSE = 0.086 ±0.000
conditional random forests: R² = 26.3% ±0.2%, RMSE = 0.075 ±0.000.
support vector machines: R² = 4.3% ±4.8%, RMSE = 0.093 ±0.009.

Std Intra-individual variation:

random forests: R² = 90.3% ±0.4%, RMSE = 0.003 ±0.000
conditional random forests: R² = 85.2% ±0.8%, RMSE = 0.004 ±0.000.
support vector machines: R² = 86.8% ±12.9%, RMSE = 0.004 ±0.003.

Figure 35. Success (using R² and RMSE indexes) in random networks in language change scenario for each type of method.

Predictor importance in language change context:

Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

Figure 36. Predictor importance in random networks in language change scenario for each type of method.

3.3.4.2 Small-world networks

Success in language emergence context:

Inter-individual variation:

random forests: R² = 90.2% ±0.5%, RMSE = 0.002 ±0.000
conditional random forests: R² = 95.3% ±0.2%, RMSE = 0.001 ±0.000.
support vector machines: R² = 92.8% ±2.2%, RMSE = 0.001 ±0.000.

Mean Intra-individual variation:

random forests: R² = 58.9% ±0.4%, RMSE = 0.024 ±0.000
conditional random forests: R² = 67.3% ±0.2%, RMSE = 0.021 ±0.000.
support vector machines: R² = 7.0% ±7.1%, RMSE = 0.070 ±0.008.

Std Intra-individual variation:

random forests: R² = 81.6% ±0.4%, RMSE = 0.004 ±0.000
conditional random forests: R² = 89.0% ±0.2%, RMSE = 0.003 ±0.000.
support vector machines: R² = 84.6% ±5.0%, RMSE = 0.009 ±0.002.

Figure 37. Success (using R² and RMSE indexes) in small-world networks in language emergence scenario for each type of method.

Predictor importance in language emergence context:

Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

Figure 38. Predictor importance in small-world networks in language emergence scenario for each type of method.

Success in language change context:

Inter-individual variation:

random forests: R² = 88.0% ±0.5%, RMSE = 0.001 ±0.000
conditional random forests: R² = 93.2% ±0.2%, RMSE = 0.001 ±0.000.
support vector machines: R² = 92.9% ±2.8%, RMSE = 0.001 ±0.000.

Mean Intra-individual variation:

random forests: R² = 4.7% ±0.7%, RMSE = 0.072 ±0.000
conditional random forests: R² = 25.4% ±0.3%, RMSE = 0.063 ±0.000.
support vector machines: R² = 6.7% ±5.8%, RMSE = 0.071 ±0.008.

Std Intra-individual variation:

random forests: R² = 82.1% ±0.3%, RMSE = 0.009 ±0.000
conditional random forests: R² = 88.7% ±0.1%, RMSE = 0.007 ±0.000.
support vector machines: R² = 84.3% ±5.1%, RMSE = 0.009 ±0.002.

Figure 39. Success in small-world networks in language change scenario for each type of method.

Predictor importance in language change context:

Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

$**Figure 40.** Predictor importance in small-world networks in language change scenario for each type of method.$

Figure 40. Predictor importance in small-world networks in language change scenario for each type of method.

3.3.4.3 Scale-free networks

Success in language emergence context:

Inter-individual variation:

random forests: R² = 37.5% ±0.4%, RMSE = 0.006 ±0.000
conditional random forests: R² = 55.8% ±0.2%, RMSE = 0.005 ±0.000.
support vector machines: R² = 37.5% ±9.3%, RMSE = 0.006 ±0.000.

Mean Intra-individual variation:

random forests: R² = 1.7% ±0.3%, RMSE = 0.016 ±0.000
conditional random forests: R² = 34.5% ±0.7%, RMSE = 0.014 ±0.000.
support vector machines: R² = 4.1% ±5.6%, RMSE = 0.016 ±0.002.

Std Intra-individual variation:

random forests: R² = 4.2% ±0.4%, RMSE = 0.010 ±0.000
conditional random forests: R² = 35.2% ±0.5%, RMSE = 0.008 ±0.000.
support vector machines: R² = 11.1% ±8.5%, RMSE = 0.009 ±0.001.

$**Figure 41.** Success (using R² and RMSE indexes) in scale-free networks in language emergence scenario for each type of method.$

Figure 41. Success (using R² and RMSE indexes) in scale-free networks in language emergence scenario for each type of method.

Predictor importance in language emergence context:

And now, let’s visualize the importance of the predictors. Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

$**Figure 42.** Predictor importance in scale-free networks in language emergence scenario for each type of method.$

Figure 42. Predictor importance in scale-free networks in language emergence scenario for each type of method.

Success in language change context:

Inter-individual variation:

random forests: R² = 36.8% ±0.4%, RMSE = 0.003 ±0.000
conditional random forests: R² = 54.5% ±0.2%, RMSE = 0.003 ±0.000.
support vector machines: R² = 39.8% ±8.4%, RMSE = 0.006 ±0.000.

Mean Intra-individual variation:

random forests: R² = 0.0% ±0.0%, RMSE = 0.047 ±0.000
conditional random forests: R² = 36.4% ±0.9%, RMSE = 0.040 ±0.000.
support vector machines: R² = 5.2% ±4.8%, RMSE = 0.016 ±0.002.

Std Intra-individual variation:

random forests: R² = 7.4% ±0.4%, RMSE = 0.019 ±0.000
conditional random forests: R² = 31.5% ±0.3%, RMSE = 0.017 ±0.000.
support vector machines: R² = 11.4% ±7.5%, RMSE = 0.009 ±0.001.

Figure 43. Success (using R² and RMSE indexes) in scale-free networks in language change scenario for each type of method.

Predictor importance in language change context:

Important: note that the metrics are printed by decreasing importance, so their order may vary between the different methods!

Figure 44. Predictor importance in scale-free networks in language change scenario for each type of method.

3.3.4.4 Summary all

Success:

In language emergence scenario:

	Inter	Mintra	Sintra
Random	91.48819	45.324598	76.260186
Small-World	90.20976	58.875546	81.583876
Scale-free	37.50218	1.723138	4.176135

In language change scenario:

	Inter	Mintra	Sintra
Random	95.86643	2.4651129	90.334545
Small-World	88.03344	4.6870283	82.062926
Scale-free	36.76559	0.0151357	7.401045

The R² values are particularly low for intra-individual variation: 35.31% on average for all types of networks in language emergence and 2.39% for language change.

The R² values are also particularly low for scale-free variation: for both language emergence and language change, it predicts 37.5% for inter-individual variation and 4.18% for standard-deviation of intra-individual variation.

The rest is well-predicted: The mean R² of inter-individual variation for language emergence and change in random and small-world network is of 91.49%, while the standard-deviation of the intra-individual variation is of 76.26%.

Predictor importance:

Plot for the paper:

Figure 45. Mean normalized predictor importance (mean of the four index including the accuracy-based and gini indexes from random forests, the index from conditional forests and the index from SVM) in random et small-world networks. We look at inter-individual variation (left panel) and standard-deviation of intra-individual variation (right panel) in language change scenario (upper panel) and language emergence scenario (lower panel). Note that we selected only the models with high success of prediction.

Plot with all data:

Figure 46. Mean normalized predictor importance (mean of the four index including the accuracy-based and gini indexes from random forests, the index from conditional forests and the index from SVM) in random, small-world et scale-fre networks. We look at inter-individual variation (left panel), mean intra-individual variation (middle panel) and standard-deviation of intra-individual variation (right panel) in language change scenario (upper panel) and language emergence scenario (lower panel)

Clearly, pathlength is the best predictor for inter-individual and standard-deviation of intra-individual variation. It is followed by clustering coefficient, and then node degree.
None of the predictors accurately predict the mean intra-individual variation.
A similar pattern as for inter-individual variation emerges for the standard-deviation of intra-individual variation.

These results only concerns the machine learning techniques. They are corrobated by the use of linear models (see part below, or go directly to Summary Modeling part for a summary).

3.3.5 Modeling

Then, we wish to observe the effect size of our different metrics using modeling. In our models, we want to investigate the relative effect of the different predictors (here, our metrics normalized) on variation (our different measures for inter- and intra- individual variability). Our measure of variation are all continuous, so we use a linear model with function lm (stats package, version 3.6.2, see here for more information).

3.3.5.1 Controling for VIF

To do so, we create a model including all metrics as predictors. Then, using this model, we first look at the VIF score (Variance Inflation Factor). In statistics, the VIF score shows the extent of multicollinearity in a regression analysis. It is “the ratio (quotient) of the variance of estimating some parameter in a model that includes multiple other terms (parameters) by the variance of a model constructed using only one terms.”.

The extent to which a VIF score is “too high” is still debated. In general, it is suggested that a VIF score greater than 5 reflects highly correlated predictors (source: here or here), but please note that there is no absolute “truth” for what is a high VIF score.

In scale-free networks:

In language emergence:

metrics	vif_score
exponent_z	2.094996
assortativity_z	2.176258
pathlength_z	3.122374
size_z	4.559353

In language change:

metrics	vif_score
exponent_z	2.232382
assortativity_z	2.052236
pathlength_z	2.936720
size_z	5.012959

We observe that no metrics have a VIF too high (above 5). Thus, we will keep all metrics in the model of scale-free networks.

In small-world networks:

In language emergence:

metrics	vif_score
clustering_z	2.212031
node_degree_z	3.581759
size_z	2.056694
assortativity_z	1.254807
pathlength_z	2.819969

In language change:

metrics	vif_score
clustering_z	2.256314
node_degree_z	3.630869
size_z	2.105092
assortativity_z	1.253142
pathlength_z	3.021626

We observe that no metrics have a VIF too high (above 5). Thus, we will keep all metrics in the model of small-world networks.

In random networks:

In language emergence:

metrics	vif_score
node_degree_z	7.368530
size_z	4.821028
clustering_z	7.776622
pathlength_z	3.560723
assortativity_z	1.380199

In language change:

metrics	vif_score
node_degree_z	7.450035
size_z	5.433643
clustering_z	7.624188
pathlength_z	3.393699
assortativity_z	1.581143

We observe that the following metrics have a VIF too high (above 5): clustering and node degree. However, the VIF score is not that much higher than 5 (7 for both metrics). To check which of these two metrics explains the most variation, we create models using only the two metrics (for both inter- and intra- individual variation):


Call:
lm(formula = inter_var * 100 ~ clustering_z + node_degree_z, 
    data = subdf_mod_ran_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.17264 -0.10718 -0.03745  0.02481  1.77137 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.051867   0.014481   3.582 0.000399 ***
clustering_z  -0.081600   0.018601  -4.387  1.6e-05 ***
node_degree_z -0.003229   0.018555  -0.174 0.861982    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2504 on 296 degrees of freedom
Multiple R-squared:  0.1006,    Adjusted R-squared:  0.0945 
F-statistic: 16.55 on 2 and 296 DF,  p-value: 1.536e-07


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + node_degree_z, 
    data = subdf_mod_ran_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-19.1812  -1.3586   0.2887   1.8521   8.1046 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)   225.4401     0.1694 1330.79   <2e-16 ***
clustering_z   -2.7810     0.2176  -12.78   <2e-16 ***
node_degree_z   3.0115     0.2171   13.87   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.929 on 296 degrees of freedom
Multiple R-squared:  0.4268,    Adjusted R-squared:  0.4229 
F-statistic: 110.2 on 2 and 296 DF,  p-value: < 2.2e-16


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + node_degree_z, 
    data = subdf_mod_ran_em)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.4323 -0.2181 -0.0823  0.0686  6.0083 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.17475    0.03273   5.339 1.86e-07 ***
clustering_z  -0.19301    0.04204  -4.591 6.54e-06 ***
node_degree_z -0.03476    0.04194  -0.829    0.408    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.566 on 296 degrees of freedom
Multiple R-squared:  0.1278,    Adjusted R-squared:  0.1219 
F-statistic: 21.69 on 2 and 296 DF,  p-value: 1.623e-09

We also did create models using only one predictor and looking at the amount of variation (R-squared) it explains, but it yield to the same conclusions: clearly, it seems that the clustering coefficient explains more variation than the node degree. Since the VIF is not that much higher than 5 and since this cutoff is arbitrary, we will first use a model containing all predictors, to ensure coherence with scale-free and small-world networks. At the end of the analysis, we will also remove node degree from the predictors, to make sure that this is does not affect our conclusions.

Summary

In scale-free and small-world networks, we keep all predictors in the model, since the VIF is lower than 5 for all metrics.
In random networks, to ensure conherence with scale-free and small-world networks, we will still analyse a model with all predictors. However, our analysis suggest that clustering coefficient explains more variation than node degree.

3.3.5.2 Controlling for other assumptions of linear models

Here, we observe the diagnostic plots for all type of networks. Since the diagnostic plots for language change are extremely similar with the diagnostic plots for language emergence scenario, we present here the plots only for language emergence.

Random Networks:

In language emergence scenario:

Figure 47. Diagnostic plots when predicting inter-individual variation (random networks, language emergence).

Figure 48. Diagnostic plots when predicting mean intra-individual variation (random networks, language emergence)

Figure 49. Diagnostic plots when predicting standard-deviation of intra-individual variation (random networks, language emergence)

The model’s assumptions are not being met. Below, we will attempt to address this issue. Given the analogous patterns observed in both language emergence and language change scenarios, we will focus on correcting the model assumptions specifically in the context of language emergence and inter-individual variation to assess its impact.

Checking linearity assumptions (residuals vs fitted, first plot)

It seems that linearity is not respected. Thus, we also check whether a quadratic model of these variables would fit better.


Call:
lm(formula = inter_var ~ size_z + clustering_z + I(clustering_z^2) + 
    node_degree_z + I(node_degree_z^2) + pathlength_z + I(pathlength_z^2) + 
    assortativity_z + I(assortativity_z^2), data = subdf_mod_ran_ch)

Residuals:
       Min         1Q     Median         3Q        Max 
-9.405e-04 -1.363e-05  3.360e-06  2.629e-05  1.777e-03 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)          -2.858e-04  3.085e-05  -9.264  < 2e-16 ***
size_z               -5.751e-05  3.422e-05  -1.680  0.09397 .  
clustering_z         -8.792e-04  7.195e-05 -12.220  < 2e-16 ***
I(clustering_z^2)    -1.685e-04  2.040e-05  -8.259 5.39e-15 ***
node_degree_z         1.362e-04  5.680e-05   2.397  0.01715 *  
I(node_degree_z^2)   -3.356e-05  1.701e-05  -1.974  0.04939 *  
pathlength_z         -1.355e-03  9.933e-05 -13.645  < 2e-16 ***
I(pathlength_z^2)     8.092e-04  1.882e-05  42.992  < 2e-16 ***
assortativity_z      -5.673e-05  1.944e-05  -2.919  0.00379 ** 
I(assortativity_z^2)  4.255e-05  6.870e-06   6.194 2.02e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.0001975 on 288 degrees of freedom
Multiple R-squared:  0.9882,    Adjusted R-squared:  0.9879 
F-statistic:  2684 on 9 and 288 DF,  p-value: < 2.2e-16

It does not change much the results, and linearity is still not respected.

Checking normality (normal Q-Q, second plot)


    Shapiro-Wilk normality test

data:  resid(model)
W = 0.45829, p-value < 2.2e-16

Significantly different from a normal distribution.

Checking for homoscedasticity (scale-location, third plot)

Non-constant Variance Score Test 
Variance formula: ~ fitted.values 
Chisquare = 2686.694, Df = 1, p = < 2.22e-16

The data has clearly heteroscedasticity. We log the output to decrease heteroscedasticity.

Non-constant Variance Score Test 
Variance formula: ~ fitted.values 
Chisquare = 2.681535, Df = 1, p = 0.10152

The data still has heteroscedasticity, but less.

Checking the influence of extreme values (Residuals versus Leverage, fourth plot).

scale-free Networks:

In language emergence scenario:

Figure 50. Diagnostic plots when predicting **inter*-individual variation (scale-free networks, language emergence)

Figure 51. Diagnostic plots when predicting mean intra-individual variation (scale-free networks, language emergence)

Figure 52. Diagnostic plots when predicting standard-deviation of intra-individual variation (scale-free networks, language emergence)

Small-world Networks:

In language emergence scenario:

Figure 53. Diagnostic plots when predicting inter-individual variation (small-world networks, language emergence)

Figure 54. Diagnostic plots when predicting mean intra-individual variation (small-world networks, language emergence)

Figure 55. Diagnostic plots when predicting standard-deviation of intra-individual variation (small-world networks, language emergence)

3.3.5.3 Methodology for the models

First, we create a linear model using these metrics as predictors and the type of variation (inter, mean intra, std intra) as the predicted variable. For easier visualizuation, please note that we multiply all types of variation by 100, in both language emergence and language change scenarios. All predictors are z-scored.

We first look at the summary of each linear models.
Then, we gather the standardized estimates in a table.
We also use step models (using stats package), which chooses a model by AIC in a stepwise Algorithm. The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. Step models show the best predictors and also indicates the most important predictors for maximizing AIC (increasing importance). In other terms, it shows how much the AIC increases when removing a predictor from the model (the higher the AIC, the worst the model).
In order to look at the variable importance, we also integrate all variables in a model, then remove each of them to evaluate the decrease in the total R² of the model.
Finally, we create models with single predictors and evaluate the total R² that these predictors explain.

3.3.5.4 Inter-individual variation

1 - Summary of the linear models.

For random networks, in language emergence:


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.38396 -0.01526  0.00626  0.03998  0.59269 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.050242   0.005232   9.603  < 2e-16 ***
clustering_z     0.182628   0.014649  12.467  < 2e-16 ***
assortativity_z -0.015529   0.006194  -2.507  0.01271 *  
pathlength_z     0.363731   0.009888  36.785  < 2e-16 ***
size_z          -0.028448   0.011487  -2.477  0.01383 *  
node_degree_z    0.039037   0.014224   2.744  0.00644 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.09046 on 293 degrees of freedom
Multiple R-squared:  0.8838,    Adjusted R-squared:  0.8818 
F-statistic: 445.7 on 5 and 293 DF,  p-value: < 2.2e-16

For random networks, in language change:


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.252490 -0.011758  0.004671  0.027376  0.292066 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.033849   0.003471   9.752  < 2e-16 ***
clustering_z     0.131623   0.009642  13.651  < 2e-16 ***
assortativity_z -0.015461   0.004422  -3.496 0.000545 ***
pathlength_z     0.250502   0.006404  39.120  < 2e-16 ***
size_z          -0.010637   0.008076  -1.317 0.188841    
node_degree_z    0.021007   0.009487   2.214 0.027587 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.0599 on 292 degrees of freedom
Multiple R-squared:  0.8902,    Adjusted R-squared:  0.8883 
F-statistic: 473.3 on 5 and 292 DF,  p-value: < 2.2e-16

For small-world networks, in language emergence:


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.46288 -0.17581 -0.01208  0.13034  1.09315 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.35133    0.01400  25.103  < 2e-16 ***
clustering_z     0.22439    0.02085  10.762  < 2e-16 ***
assortativity_z  0.08090    0.01570   5.152 4.73e-07 ***
pathlength_z     0.68205    0.02354  28.972  < 2e-16 ***
size_z          -0.11873    0.02010  -5.905 9.70e-09 ***
node_degree_z    0.04566    0.02653   1.721   0.0863 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2424 on 294 degrees of freedom
Multiple R-squared:  0.8479,    Adjusted R-squared:  0.8453 
F-statistic: 327.8 on 5 and 294 DF,  p-value: < 2.2e-16

For small-world networks, in language change:


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.30734 -0.11059 -0.00921  0.07822  1.00512 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.236805   0.009445  25.072  < 2e-16 ***
clustering_z     0.140855   0.014211   9.912  < 2e-16 ***
assortativity_z  0.044994   0.010591   4.248 2.89e-05 ***
pathlength_z     0.420955   0.016445  25.597  < 2e-16 ***
size_z          -0.074567   0.013726  -5.432 1.17e-07 ***
node_degree_z    0.023018   0.018027   1.277    0.203    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1636 on 294 degrees of freedom
Multiple R-squared:  0.8197,    Adjusted R-squared:  0.8167 
F-statistic: 267.4 on 5 and 294 DF,  p-value: < 2.2e-16

For scale-free networks, in language emergence:


Call:
lm(formula = inter_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.47445 -0.33898 -0.01628  0.31736  1.61598 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      3.30238    0.03205 103.045  < 2e-16 ***
exponent_z      -0.07074    0.04646  -1.523  0.12894    
assortativity_z -0.01776    0.04736  -0.375  0.70791    
pathlength_z     0.32245    0.05672   5.685 3.15e-08 ***
size_z           0.18763    0.06854   2.737  0.00657 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5551 on 295 degrees of freedom
Multiple R-squared:  0.3867,    Adjusted R-squared:  0.3784 
F-statistic:  46.5 on 4 and 295 DF,  p-value: < 2.2e-16

For scale-free networks, in language change:


Call:
lm(formula = inter_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_ch)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.76479 -0.21379 -0.01904  0.18163  1.69505 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)      2.035471   0.019081 106.673  < 2e-16 ***
exponent_z      -0.002603   0.028557  -0.091   0.9274    
assortativity_z  0.027438   0.027381   1.002   0.3171    
pathlength_z     0.187230   0.032754   5.716 2.67e-08 ***
size_z           0.072220   0.042794   1.688   0.0925 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3305 on 295 degrees of freedom
Multiple R-squared:  0.3951,    Adjusted R-squared:  0.3869 
F-statistic: 48.18 on 4 and 295 DF,  p-value: < 2.2e-16

2 - Summary of the standardized estimates of the linear models.

For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.

In language emergence:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	0.05	0.35	3.30
assortativity	0.02	0.08	0.02
clustering	0.18	0.22	NA
exponent	NA	NA	0.07
node degree	0.04	0.05	NA
pathlength	0.36	0.68	0.32
size	0.03	0.12	0.19

In language change:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	0.03	0.24	2.04
assortativity	0.02	0.04	0.03
clustering	0.13	0.14	NA
exponent	NA	NA	0.00
node degree	0.02	0.02	NA
pathlength	0.25	0.42	0.19
size	0.01	0.07	0.07

3 - Step models (looking at AIC).

For random networks, in language emergence:

Start:  AIC=-1430.97
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq     RSS     AIC
<none>                          2.3976 -1431.0
- size_z           1    0.0502  2.4478 -1426.8
- assortativity_z  1    0.0514  2.4490 -1426.6
- node_degree_z    1    0.0616  2.4592 -1425.4
- clustering_z     1    1.2718  3.6694 -1305.7
- pathlength_z     1   11.0723 13.4699  -916.9


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
        0.05024          0.18263         -0.01553          0.36373  
         size_z    node_degree_z  
       -0.02845          0.03904

For random networks, in language change:

Start:  AIC=-1671.8
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS     AIC
- size_z           1    0.0062 1.0541 -1672.0
<none>                         1.0479 -1671.8
- node_degree_z    1    0.0176 1.0654 -1668.8
- assortativity_z  1    0.0439 1.0917 -1661.6
- clustering_z     1    0.6688 1.7166 -1526.7
- pathlength_z     1    5.4917 6.5395 -1128.1

Step:  AIC=-1672.04
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    node_degree_z

                  Df Sum of Sq    RSS     AIC
<none>                         1.0541 -1672.0
- node_degree_z    1    0.0158 1.0699 -1669.6
- assortativity_z  1    0.0648 1.1189 -1656.3
- clustering_z     1    1.5305 2.5846 -1406.8
- pathlength_z     1    6.1628 7.2168 -1100.8


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + node_degree_z, data = subdf_mod_ran_ch)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
        0.03380          0.14061         -0.01754          0.25304  
  node_degree_z  
        0.01031

For small-world networks, in language emergence:

Start:  AIC=-844.33
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS     AIC
<none>                         17.276 -844.33
- node_degree_z    1     0.174 17.450 -843.33
- assortativity_z  1     1.560 18.836 -820.40
- size_z           1     2.049 19.326 -812.70
- clustering_z     1     6.806 24.082 -746.70
- pathlength_z     1    49.324 66.600 -441.52


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
        0.35133          0.22439          0.08090          0.68205  
         size_z    node_degree_z  
       -0.11873          0.04566

For small-world networks, in language change:

Start:  AIC=-1080.3
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq     RSS      AIC
- node_degree_z    1    0.0436  7.9116 -1080.64
<none>                          7.8679 -1080.30
- assortativity_z  1    0.4830  8.3510 -1064.42
- size_z           1    0.7897  8.6577 -1053.60
- clustering_z     1    2.6292 10.4971  -995.80
- pathlength_z     1   17.5349 25.4028  -730.68

Step:  AIC=-1080.64
inter_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS      AIC
<none>                          7.912 -1080.64
- assortativity_z  1     0.462  8.373 -1065.62
- size_z           1     0.889  8.800 -1050.70
- clustering_z     1     4.906 12.817  -937.90
- pathlength_z     1    34.134 42.046  -581.51


Call:
lm(formula = inter_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sw_ch)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
        0.23681          0.15198          0.04382          0.40577  
         size_z  
       -0.06439

For scale-free networks, in language emergence:

Start:  AIC=-348.22
inter_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq     RSS     AIC
- assortativity_z  1    0.0433  90.938 -350.08
<none>                          90.895 -348.22
- exponent_z       1    0.7143  91.609 -347.88
- size_z           1    2.3088  93.204 -342.70
- pathlength_z     1    9.9567 100.851 -319.04

Step:  AIC=-350.08
inter_var * 100 ~ exponent_z + pathlength_z + size_z

               Df Sum of Sq     RSS     AIC
<none>                       90.938 -350.08
- exponent_z    1    0.8760  91.814 -349.20
- size_z        1    2.2863  93.224 -344.63
- pathlength_z  1   10.0827 101.021 -320.54


Call:
lm(formula = inter_var * 100 ~ exponent_z + pathlength_z + size_z, 
    data = subdf_mod_sf_em)

Coefficients:
 (Intercept)    exponent_z  pathlength_z        size_z  
     3.30238      -0.07544       0.31833       0.18174

For scale-free networks, in language change:

Start:  AIC=-659.33
inter_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS     AIC
- exponent_z       1    0.0009 32.224 -661.32
- assortativity_z  1    0.1097 32.333 -660.31
<none>                         32.223 -659.33
- size_z           1    0.3111 32.534 -658.45
- pathlength_z     1    3.5691 35.792 -629.82

Step:  AIC=-661.32
inter_var * 100 ~ assortativity_z + pathlength_z + size_z

                  Df Sum of Sq    RSS     AIC
- assortativity_z  1    0.1108 32.335 -662.29
<none>                         32.224 -661.32
- size_z           1    0.4046 32.628 -659.58
- pathlength_z     1    3.8345 36.058 -629.59

Step:  AIC=-662.29
inter_var * 100 ~ pathlength_z + size_z

               Df Sum of Sq    RSS     AIC
<none>                      32.335 -662.29
- size_z        1    0.8466 33.181 -656.54
- pathlength_z  1    3.8690 36.204 -630.39


Call:
lm(formula = inter_var * 100 ~ pathlength_z + size_z, data = subdf_mod_sf_ch)

Coefficients:
 (Intercept)  pathlength_z        size_z  
      2.0355        0.1888        0.0883

4 - Drop in R² when removing predictors

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.0616377	0.0599168	NA
node_degree	0.0029870	0.0015321	NA
assortativity	0.0024929	0.0137317	0.0002924
pathlength	0.5366329	0.4342481	0.0671823
size	0.0024325	0.0180421	0.0155788
exponent	NA	NA	0.0048196

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.0701024	0.0602336	NA
node_degree	0.0018442	0.0009996	NA
assortativity	0.0045983	0.0110663	0.0020589
pathlength	0.5756553	0.4017201	0.0669968
size	0.0006525	0.0180929	0.0058397
exponent	NA	NA	0.0000170

5 - R² explained by each predictor

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.0974584	-0.0014516	NA
node_degree	0.0388764	0.1682322	NA
assortativity	0.1480515	-0.0032237	0.1495115
pathlength	0.6182816	0.6401498	0.3687975
size	0.0413732	0.0136092	0.3018147
exponent	NA	NA	0.0824252

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.0998072	-0.0027987	NA
node_degree	0.0398155	0.1896333	NA
assortativity	0.1463084	0.0048742	0.1841091
pathlength	0.6218682	0.6223793	0.3750539
size	0.0418832	0.0092948	0.3181298
exponent	NA	NA	0.1261277

3.3.5.5 Mean intra-individual variation

We apply here exactly the same steps, but using the mean intra-individual variation as the predicted variable. Please note that here too, for better visualization, we multiply the mean intra-individual variation by 100.

1 - Summary of the linear models.

For random networks, in language emergence:


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-19.9060  -1.2994   0.2065   1.7039   7.5610 

Coefficients:
                Estimate Std. Error  t value Pr(>|t|)    
(Intercept)     225.4411     0.1636 1377.875  < 2e-16 ***
clustering_z     -1.5649     0.4581   -3.416 0.000726 ***
assortativity_z   0.3292     0.1937    1.700 0.090252 .  
pathlength_z      0.1962     0.3092    0.635 0.526233    
size_z            1.4270     0.3592    3.972 8.96e-05 ***
node_degree_z     1.2852     0.4448    2.889 0.004151 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.829 on 293 degrees of freedom
Multiple R-squared:  0.4707,    Adjusted R-squared:  0.4617 
F-statistic: 52.12 on 5 and 293 DF,  p-value: < 2.2e-16

For random networks, in language change:


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)

Residuals:
    Min      1Q  Median      3Q     Max 
-35.987  -4.759   0.314   4.808  25.321 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     195.4888     0.4879 400.647   <2e-16 ***
clustering_z     -2.8852     1.3554  -2.129   0.0341 *  
assortativity_z   1.0242     0.6216   1.648   0.1005    
pathlength_z      0.1762     0.9002   0.196   0.8450    
size_z           -0.5688     1.1353  -0.501   0.6167    
node_degree_z     2.9409     1.3337   2.205   0.0282 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 8.421 on 292 degrees of freedom
Multiple R-squared:  0.08716,   Adjusted R-squared:  0.07153 
F-statistic: 5.576 on 5 and 292 DF,  p-value: 6.343e-05

For small-world networks, in language emergence:


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-12.7624  -0.8564   0.0747   1.2064   7.2658 

Coefficients:
                Estimate Std. Error  t value Pr(>|t|)    
(Intercept)     226.0901     0.1415 1597.391  < 2e-16 ***
clustering_z     -2.0967     0.2109   -9.944  < 2e-16 ***
assortativity_z   0.2081     0.1588    1.311    0.191    
pathlength_z     -0.3928     0.2381   -1.650    0.100    
size_z            1.3657     0.2033    6.717 9.59e-11 ***
node_degree_z    -0.1165     0.2683   -0.434    0.665    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.451 on 294 degrees of freedom
Multiple R-squared:  0.5619,    Adjusted R-squared:  0.5545 
F-statistic: 75.43 on 5 and 294 DF,  p-value: < 2.2e-16

For small-world networks, in language change:


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)

Residuals:
     Min       1Q   Median       3Q      Max 
-28.4825  -3.5119   0.3649   4.2764  20.2090 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     195.7702     0.4018 487.277  < 2e-16 ***
clustering_z     -1.6216     0.6045  -2.683  0.00772 ** 
assortativity_z   0.3445     0.4505   0.765  0.44509    
pathlength_z     -0.8917     0.6995  -1.275  0.20342    
size_z            1.2261     0.5839   2.100  0.03659 *  
node_degree_z    -1.2524     0.7668  -1.633  0.10348    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.959 on 294 degrees of freedom
Multiple R-squared:  0.1209,    Adjusted R-squared:  0.106 
F-statistic:  8.09 on 5 and 294 DF,  p-value: 3.618e-07

For scale-free networks, in language emergence:


Call:
lm(formula = mean_intra_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_em)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.4006 -0.8030  0.0626  0.8368  3.8055 

Coefficients:
                 Estimate Std. Error  t value Pr(>|t|)    
(Intercept)     221.71443    0.09106 2434.878   <2e-16 ***
exponent_z        0.17717    0.13202    1.342   0.1806    
assortativity_z   0.34523    0.13455    2.566   0.0108 *  
pathlength_z      0.05753    0.16117    0.357   0.7214    
size_z           -0.17637    0.19476   -0.906   0.3659    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.577 on 295 degrees of freedom
Multiple R-squared:  0.05738,   Adjusted R-squared:  0.0446 
F-statistic: 4.489 on 4 and 295 DF,  p-value: 0.001552

For scale-free networks, in language change:


Call:
lm(formula = mean_intra_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_ch)

Residuals:
     Min       1Q   Median       3Q      Max 
-14.2636  -2.5661   0.4168   2.8622  16.3981 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)     192.97293    0.26333 732.805   <2e-16 ***
exponent_z        0.44996    0.39411   1.142    0.255    
assortativity_z   0.07738    0.37787   0.205    0.838    
pathlength_z      0.04983    0.45203   0.110    0.912    
size_z           -0.14192    0.59058  -0.240    0.810    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.561 on 295 degrees of freedom
Multiple R-squared:  0.008748,  Adjusted R-squared:  -0.004693 
F-statistic: 0.6508 on 4 and 295 DF,  p-value: 0.6267

2 - Summary of the standardized estimates of the linear models.

For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.

In language emergence:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	225.44	226.09	221.71
assortativity	0.33	0.21	0.35
clustering	1.56	2.10	NA
exponent	NA	NA	0.18
node degree	1.29	0.12	NA
pathlength	0.20	0.39	0.06
size	1.43	1.37	0.18

In language change:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	195.49	195.77	192.97
assortativity	1.02	0.34	0.08
clustering	2.89	1.62	NA
exponent	NA	NA	0.45
node degree	2.94	1.25	NA
pathlength	0.18	0.89	0.05
size	0.57	1.23	0.14

3 - Step models (looking at AIC).

For random networks, in language emergence:

Start:  AIC=627.81
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS    AIC
- pathlength_z     1     3.222 2348.1 626.22
<none>                         2344.9 627.81
- assortativity_z  1    23.120 2368.0 628.74
- node_degree_z    1    66.804 2411.7 634.21
- clustering_z     1    93.381 2438.3 637.48
- size_z           1   126.290 2471.2 641.49

Step:  AIC=626.22
mean_intra_var * 100 ~ clustering_z + assortativity_z + size_z + 
    node_degree_z

                  Df Sum of Sq    RSS    AIC
<none>                         2348.1 626.22
- assortativity_z  1    20.005 2368.1 626.76
- node_degree_z    1    87.482 2435.6 635.16
- size_z           1   126.851 2475.0 639.95
- clustering_z     1   265.357 2613.5 656.23


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    size_z + node_degree_z, data = subdf_mod_ran_em)

Coefficients:
    (Intercept)     clustering_z  assortativity_z           size_z  
       225.4416          -1.7795           0.2913           1.3533  
  node_degree_z  
         1.3819

For random networks, in language change:

Start:  AIC=1275.86
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq   RSS    AIC
- pathlength_z     1      2.72 20710 1273.9
- size_z           1     17.80 20725 1274.1
<none>                         20707 1275.9
- assortativity_z  1    192.52 20900 1276.6
- clustering_z     1    321.34 21029 1278.5
- node_degree_z    1    344.83 21052 1278.8

Step:  AIC=1273.9
mean_intra_var * 100 ~ clustering_z + assortativity_z + size_z + 
    node_degree_z

                  Df Sum of Sq   RSS    AIC
- size_z           1     24.46 20734 1272.2
<none>                         20710 1273.9
- assortativity_z  1    190.87 20901 1274.6
- node_degree_z    1    406.48 21117 1277.7
- clustering_z     1    750.65 21461 1282.5

Step:  AIC=1272.25
mean_intra_var * 100 ~ clustering_z + assortativity_z + node_degree_z

                  Df Sum of Sq   RSS    AIC
<none>                         20734 1272.2
- assortativity_z  1    168.19 20903 1272.7
- node_degree_z    1    873.10 21608 1282.5
- clustering_z     1   1245.96 21980 1287.6


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + assortativity_z + 
    node_degree_z, data = subdf_mod_ran_ch)

Coefficients:
    (Intercept)     clustering_z  assortativity_z    node_degree_z  
       195.4880          -2.6645           0.8417           2.4009

For small-world networks, in language emergence:

Start:  AIC=543.96
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS    AIC
- node_degree_z    1      1.13 1768.0 542.15
- assortativity_z  1     10.32 1777.2 543.71
<none>                         1766.9 543.96
- pathlength_z     1     16.36 1783.2 544.72
- size_z           1    271.14 2038.0 584.79
- clustering_z     1    594.24 2361.1 628.93

Step:  AIC=542.15
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS    AIC
- assortativity_z  1      9.93 1778.0 541.83
<none>                         1768.0 542.15
- pathlength_z     1     22.11 1790.1 543.88
- size_z           1    359.76 2127.8 595.72
- clustering_z     1    963.24 2731.3 670.62

Step:  AIC=541.83
mean_intra_var * 100 ~ clustering_z + pathlength_z + size_z

               Df Sum of Sq    RSS    AIC
<none>                      1778.0 541.83
- pathlength_z  1     29.19 1807.2 544.72
- size_z        1    464.02 2242.0 609.40
- clustering_z  1    998.80 2776.8 673.58


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + pathlength_z + 
    size_z, data = subdf_mod_sw_em)

Coefficients:
 (Intercept)  clustering_z  pathlength_z        size_z  
    226.0901       -2.0836       -0.3596        1.3962

For small-world networks, in language change:

Start:  AIC=1169.94
mean_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq   RSS    AIC
- assortativity_z  1     28.31 14265 1168.5
- pathlength_z     1     78.68 14315 1169.6
<none>                         14237 1169.9
- node_degree_z    1    129.17 14366 1170.7
- size_z           1    213.53 14450 1172.4
- clustering_z     1    348.48 14585 1175.2

Step:  AIC=1168.53
mean_intra_var * 100 ~ clustering_z + pathlength_z + size_z + 
    node_degree_z

                Df Sum of Sq   RSS    AIC
<none>                       14265 1168.5
- pathlength_z   1    110.42 14375 1168.8
- node_degree_z  1    140.95 14406 1169.5
- size_z         1    292.89 14558 1172.6
- clustering_z   1    320.54 14586 1173.2


Call:
lm(formula = mean_intra_var * 100 ~ clustering_z + pathlength_z + 
    size_z + node_degree_z, data = subdf_mod_sw_ch)

Coefficients:
  (Intercept)   clustering_z   pathlength_z         size_z  node_degree_z  
      195.770         -1.505         -1.024          1.365         -1.303

For scale-free networks, in language emergence:

Start:  AIC=278.34
mean_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS    AIC
- pathlength_z     1    0.3170 734.12 276.46
- size_z           1    2.0400 735.84 277.17
- exponent_z       1    4.4797 738.28 278.16
<none>                         733.80 278.34
- assortativity_z  1   16.3748 750.17 282.96

Step:  AIC=276.47
mean_intra_var * 100 ~ exponent_z + assortativity_z + size_z

                  Df Sum of Sq    RSS    AIC
- size_z           1    2.0071 736.12 275.28
- exponent_z       1    4.1829 738.30 276.17
<none>                         734.12 276.46
- assortativity_z  1   17.9453 752.06 281.71

Step:  AIC=275.28
mean_intra_var * 100 ~ exponent_z + assortativity_z

                  Df Sum of Sq    RSS    AIC
- exponent_z       1    2.4603 738.58 274.29
<none>                         736.12 275.28
- assortativity_z  1   16.5437 752.67 279.95

Step:  AIC=274.29
mean_intra_var * 100 ~ assortativity_z

                  Df Sum of Sq    RSS    AIC
<none>                         738.58 274.29
- assortativity_z  1    39.882 778.46 288.06


Call:
lm(formula = mean_intra_var * 100 ~ assortativity_z, data = subdf_mod_sf_em)

Coefficients:
    (Intercept)  assortativity_z  
       221.7144           0.3652

For scale-free networks, in language change:

Start:  AIC=915.5
mean_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS    AIC
- pathlength_z     1    0.2528 6137.3 913.51
- assortativity_z  1    0.8725 6137.9 913.54
- size_z           1    1.2013 6138.2 913.55
- exponent_z       1   27.1172 6164.2 914.82
<none>                         6137.0 915.50

Step:  AIC=913.51
mean_intra_var * 100 ~ exponent_z + assortativity_z + size_z

                  Df Sum of Sq    RSS    AIC
- assortativity_z  1    0.9543 6138.3 911.55
- size_z           1    1.0751 6138.4 911.56
- exponent_z       1   27.5329 6164.8 912.85
<none>                         6137.3 913.51

Step:  AIC=911.55
mean_intra_var * 100 ~ exponent_z + size_z

             Df Sum of Sq    RSS    AIC
- size_z      1    0.4169 6138.7 909.57
- exponent_z  1   31.1564 6169.4 911.07
<none>                    6138.3 911.55

Step:  AIC=909.57
mean_intra_var * 100 ~ exponent_z

             Df Sum of Sq    RSS    AIC
<none>                    6138.7 909.57
- exponent_z  1    52.535 6191.2 910.13


Call:
lm(formula = mean_intra_var * 100 ~ exponent_z, data = subdf_mod_sf_ch)

Coefficients:
(Intercept)   exponent_z  
   192.9729       0.4192

4 - Drop in R² when removing predictors

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.0210765	0.1473304	NA
node_degree	0.0150780	0.0002807	NA
assortativity	0.0052184	0.0025592	0.0210348
pathlength	0.0007273	0.0040550	0.0004072
size	0.0285042	0.0672249	0.0026205
exponent	NA	NA	0.0057545

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.0141656	0.0215168	NA
node_degree	0.0152013	0.0079759	NA
assortativity	0.0084867	0.0017482	0.0001409
pathlength	0.0001197	0.0048583	0.0000408
size	0.0007847	0.0131846	0.0001940
exponent	NA	NA	0.0043799

5 - R² explained by each predictor

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.0508113	0.4421897	NA
node_degree	0.1074698	0.0737294	NA
assortativity	0.0964938	-0.0029023	0.0480474
pathlength	0.0149308	0.0825762	0.0152155
size	0.3904400	0.3022928	0.0178461
exponent	NA	NA	0.0298955

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.0109673	0.0977439	NA
node_degree	0.0120052	0.0382061	NA
assortativity	0.0237402	-0.0031980	0.0003619
pathlength	0.0041205	0.0176431	-0.0016285
size	0.0467455	0.0396394	0.0001765
exponent	NA	NA	0.0051582

3.3.5.6 Std intra-individual variation

We apply here exactly the same steps, but using the std intra-individual variation as the predicted variable. Please note that here too, for better visualizuation, we multiply the std intra-individual variation by 100.

1 - Summary of the linear models.

For random networks, in language emergence:


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_em)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.4066 -0.0261  0.0054  0.0445  3.5336 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.17183    0.01723   9.974  < 2e-16 ***
clustering_z     0.28655    0.04824   5.940 8.05e-09 ***
assortativity_z  0.01291    0.02040   0.633  0.52712    
pathlength_z     0.73946    0.03256  22.710  < 2e-16 ***
size_z          -0.15570    0.03783  -4.116 5.01e-05 ***
node_degree_z    0.13964    0.04684   2.981  0.00311 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2979 on 293 degrees of freedom
Multiple R-squared:  0.7608,    Adjusted R-squared:  0.7568 
F-statistic: 186.4 on 5 and 293 DF,  p-value: < 2.2e-16

For random networks, in language change:


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.6454 -0.0497  0.0046  0.0599  3.2076 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.39832    0.01712  23.268  < 2e-16 ***
clustering_z     0.39675    0.04755   8.344 2.90e-15 ***
assortativity_z -0.14323    0.02181  -6.567 2.34e-10 ***
pathlength_z     1.20515    0.03158  38.160  < 2e-16 ***
size_z          -0.15759    0.03983  -3.956 9.56e-05 ***
node_degree_z    0.19657    0.04679   4.201 3.53e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2954 on 292 degrees of freedom
Multiple R-squared:  0.9119,    Adjusted R-squared:  0.9104 
F-statistic: 604.4 on 5 and 292 DF,  p-value: < 2.2e-16

For small-world networks, in language emergence:


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_em)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.78311 -0.24866 -0.02859  0.17983  2.13887 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.660748   0.022323  29.599  < 2e-16 ***
clustering_z     0.236637   0.033257   7.115 8.55e-12 ***
assortativity_z  0.121258   0.025048   4.841 2.09e-06 ***
pathlength_z     0.861297   0.037550  22.937  < 2e-16 ***
size_z          -0.243309   0.032068  -7.587 4.33e-13 ***
node_degree_z   -0.005828   0.042319  -0.138    0.891    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3867 on 294 degrees of freedom
Multiple R-squared:  0.788, Adjusted R-squared:  0.7844 
F-statistic: 218.5 on 5 and 294 DF,  p-value: < 2.2e-16

For small-world networks, in language change:


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.0754 -0.6464 -0.1776  0.3638  5.5210 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      2.12077    0.05996  35.372  < 2e-16 ***
clustering_z     0.68179    0.09021   7.558 5.24e-13 ***
assortativity_z  0.10574    0.06723   1.573 0.116824    
pathlength_z     2.01130    0.10439  19.266  < 2e-16 ***
size_z          -0.29570    0.08713  -3.394 0.000785 ***
node_degree_z   -0.32307    0.11444  -2.823 0.005080 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.038 on 294 degrees of freedom
Multiple R-squared:  0.7779,    Adjusted R-squared:  0.7741 
F-statistic: 205.9 on 5 and 294 DF,  p-value: < 2.2e-16

For scale-free networks, in language emergence:


Call:
lm(formula = std_intra_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_em)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.2945 -0.5700 -0.1046  0.4263  4.7065 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      3.97639    0.05451  72.952   <2e-16 ***
exponent_z      -0.07701    0.07903  -0.975    0.331    
assortativity_z  0.11644    0.08054   1.446    0.149    
pathlength_z     0.09859    0.09648   1.022    0.308    
size_z           0.12200    0.11658   1.046    0.296    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9441 on 295 degrees of freedom
Multiple R-squared:  0.07062,   Adjusted R-squared:  0.05802 
F-statistic: 5.604 on 4 and 295 DF,  p-value: 0.0002323

For scale-free networks, in language change:


Call:
lm(formula = std_intra_var * 100 ~ exponent_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sf_ch)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.6536 -1.2839 -0.0560  0.9913  6.5168 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      8.71293    0.10685  81.541   <2e-16 ***
exponent_z       0.08114    0.15992   0.507   0.6123    
assortativity_z  0.22253    0.15333   1.451   0.1478    
pathlength_z     0.42153    0.18342   2.298   0.0222 *  
size_z           0.14277    0.23964   0.596   0.5518    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.851 on 295 degrees of freedom
Multiple R-squared:  0.1445,    Adjusted R-squared:  0.1329 
F-statistic: 12.45 on 4 and 295 DF,  p-value: 2.253e-09

2 - Summary of the standardized estimates of the linear models.

For easier visualization, the following table gathers the effect size, in absolute value, rounded and ordered from the highest to the lowest, for each network type.

In language emergence:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	0.17	0.66	3.98
assortativity	0.01	0.12	0.12
clustering	0.29	0.24	NA
exponent	NA	NA	0.08
node degree	0.14	0.01	NA
pathlength	0.74	0.86	0.10
size	0.16	0.24	0.12

In language change:

metrics	Random	SmallWorld	ScaleFree
(Intercept)	0.40	2.12	8.71
assortativity	0.14	0.11	0.22
clustering	0.40	0.68	NA
exponent	NA	NA	0.08
node degree	0.20	0.32	NA
pathlength	1.21	2.01	0.42
size	0.16	0.30	0.14

3 - Step models (looking at AIC).

For random networks, in language emergence:

Start:  AIC=-718.29
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS     AIC
- assortativity_z  1     0.036 26.033 -719.89
<none>                         25.997 -718.29
- node_degree_z    1     0.789 26.786 -711.36
- size_z           1     1.503 27.501 -703.49
- clustering_z     1     3.131 29.128 -686.29
- pathlength_z     1    45.762 71.759 -416.71

Step:  AIC=-719.89
std_intra_var * 100 ~ clustering_z + pathlength_z + size_z + 
    node_degree_z

                Df Sum of Sq    RSS     AIC
<none>                       26.033 -719.89
- node_degree_z  1     0.808 26.841 -712.75
- size_z         1     1.468 27.501 -705.49
- clustering_z   1     3.110 29.143 -688.14
- pathlength_z   1    49.713 75.746 -402.54


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + pathlength_z + 
    size_z + node_degree_z, data = subdf_mod_ran_em)

Coefficients:
  (Intercept)   clustering_z   pathlength_z         size_z  node_degree_z  
       0.1718         0.2812         0.7331        -0.1519         0.1411

For random networks, in language change:

Start:  AIC=-720.75
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq     RSS     AIC
<none>                          25.487 -720.75
- size_z           1     1.366  26.854 -707.19
- node_degree_z    1     1.541  27.028 -705.27
- assortativity_z  1     3.765  29.252 -681.70
- clustering_z     1     6.076  31.564 -659.03
- pathlength_z     1   127.107 152.594 -189.46


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_ran_ch)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
         0.3983           0.3968          -0.1432           1.2052  
         size_z    node_degree_z  
        -0.1576           0.1966

For small-world networks, in language emergence:

Start:  AIC=-564.2
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq     RSS     AIC
- node_degree_z    1     0.003  43.956 -566.18
<none>                          43.953 -564.20
- assortativity_z  1     3.504  47.457 -543.19
- clustering_z     1     7.569  51.522 -518.53
- size_z           1     8.606  52.559 -512.55
- pathlength_z     1    78.656 122.609 -258.43

Step:  AIC=-566.18
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq     RSS     AIC
<none>                          43.956 -566.18
- assortativity_z  1     3.505  47.461 -545.16
- clustering_z     1    11.393  55.349 -499.04
- size_z           1    12.515  56.471 -493.02
- pathlength_z     1   162.778 206.734 -103.70


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z, data = subdf_mod_sw_em)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
         0.6607           0.2339           0.1210           0.8650  
         size_z  
        -0.2457

For small-world networks, in language change:

Start:  AIC=28.59
std_intra_var * 100 ~ clustering_z + assortativity_z + pathlength_z + 
    size_z + node_degree_z

                  Df Sum of Sq    RSS     AIC
<none>                         317.05  28.585
- assortativity_z  1      2.67 319.72  29.099
- node_degree_z    1      8.60 325.65  34.610
- size_z           1     12.42 329.47  38.112
- clustering_z     1     61.60 378.65  79.850
- pathlength_z     1    400.30 717.35 271.534


Call:
lm(formula = std_intra_var * 100 ~ clustering_z + assortativity_z + 
    pathlength_z + size_z + node_degree_z, data = subdf_mod_sw_ch)

Coefficients:
    (Intercept)     clustering_z  assortativity_z     pathlength_z  
         2.1208           0.6818           0.1057           2.0113  
         size_z    node_degree_z  
        -0.2957          -0.3231

For scale-free networks, in language emergence:

Start:  AIC=-29.56
std_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS     AIC
- exponent_z       1   0.84646 263.78 -30.596
- pathlength_z     1   0.93082 263.87 -30.500
- size_z           1   0.97605 263.91 -30.448
<none>                         262.94 -29.560
- assortativity_z  1   1.86287 264.80 -29.442

Step:  AIC=-30.6
std_intra_var * 100 ~ assortativity_z + pathlength_z + size_z

                  Df Sum of Sq    RSS     AIC
- size_z           1   0.39653 264.18 -32.145
- assortativity_z  1   1.34494 265.13 -31.070
- pathlength_z     1   1.37558 265.16 -31.035
<none>                         263.78 -30.596

Step:  AIC=-32.15
std_intra_var * 100 ~ assortativity_z + pathlength_z

                  Df Sum of Sq    RSS     AIC
<none>                         264.18 -32.145
- assortativity_z  1    2.4566 266.64 -31.368
- pathlength_z     1    4.5688 268.75 -29.001


Call:
lm(formula = std_intra_var * 100 ~ assortativity_z + pathlength_z, 
    data = subdf_mod_sf_em)

Coefficients:
    (Intercept)  assortativity_z     pathlength_z  
         3.9764           0.1169           0.1595

For scale-free networks, in language change:

Start:  AIC=374.31
std_intra_var * 100 ~ exponent_z + assortativity_z + pathlength_z + 
    size_z

                  Df Sum of Sq    RSS    AIC
- exponent_z       1    0.8817 1011.3 372.57
- size_z           1    1.2157 1011.7 372.67
<none>                         1010.5 374.31
- assortativity_z  1    7.2145 1017.7 374.45
- pathlength_z     1   18.0913 1028.5 377.63

Step:  AIC=372.57
std_intra_var * 100 ~ assortativity_z + pathlength_z + size_z

                  Df Sum of Sq    RSS    AIC
- size_z           1    3.5005 1014.8 371.61
<none>                         1011.3 372.57
- assortativity_z  1    8.8381 1020.2 373.18
- pathlength_z     1   17.2246 1028.6 375.64

Step:  AIC=371.61
std_intra_var * 100 ~ assortativity_z + pathlength_z

                  Df Sum of Sq    RSS    AIC
<none>                         1014.8 371.61
- assortativity_z  1    20.061 1034.9 375.48
- pathlength_z     1    55.014 1069.8 385.45


Call:
lm(formula = std_intra_var * 100 ~ assortativity_z + pathlength_z, 
    data = subdf_mod_sf_ch)

Coefficients:
    (Intercept)  assortativity_z     pathlength_z  
         8.7129           0.3145           0.5208

4 - Drop in R² when removing predictors

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.0288024	0.0365109	NA
node_degree	0.0072549	0.0000137	NA
assortativity	0.0003272	0.0169003	0.0065845
pathlength	0.4209669	0.3794113	0.0032901
size	0.0138290	0.0415141	0.0034500
exponent	NA	NA	0.0029919

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.0210065	0.0431583	NA
node_degree	0.0053256	0.0060219	NA
assortativity	0.0130148	0.0018692	0.0061083
pathlength	0.4394101	0.2804588	0.0153175
size	0.0047232	0.0087014	0.0010293
exponent	NA	NA	0.0007465

5 - R² explained by each predictor

Language emergence:

term	Random	SmallWorld	ScaleFree
clustering	0.1228450	-0.0031910	NA
node_degree	0.0625701	0.2353555	NA
assortativity	0.0986972	-0.0024494	0.0468897
pathlength	0.5759031	0.6008074	0.0543806
size	0.0592300	-0.0021660	0.0531710
exponent	NA	NA	0.0134618

Language change:

term	Random	SmallWorld	ScaleFree
clustering	0.2162100	0.0106864	NA
node_degree	0.1085096	0.3320156	NA
assortativity	0.2084149	0.0306273	0.0911472
pathlength	0.7483989	0.6690141	0.1208406
size	0.0678856	0.0096082	0.1181864
exponent	NA	NA	0.0629547

3.3.5.7 Summary Modeling part

The models strongly suggest that pathlength is the most importance predictor predicting inter-individual variation, followed by the clustering coefficient.
The metrics do not predict well the mean intra-individual variation (R² too low).
For the standard-deviation of intra-individual, we also do find pathlength as the main predictor, followed by the clustering coefficient.

These results corrobate the one obtained with the different machine learning techniques. The only difference is that here, node degree is rarely an important predictor. This could be due to the fact that linear models do not handle well the high auto-correlation between the metrics (and node degree is strongly correlated to other metrics), while other machine learning are supposed to.

3.4 DATASET 3: inter-individual variation within the network

In this part of the paper, we investigate the influence of local network heterogeneity on language change.

To do so, the agents do not have the same initial Dirichlet distribution, but here, 20% of them are biased. A biased agents is an agent who has an initial Dirichlet distribution giving preference to utterance $u_7$. These biased agents occupy nodes in the network based on their local metrics: for example, we bias the 20% agents with the highest or lowest node degree. These biased agents contrast with the unbiased agents, which either have no preference, either favor utterance $u_4$ (both conditions for the initial language exposure variable: see Note on Initial language exposure terminology).

With this, we compare the effects of different local network metrics (see Table below), and we also compare these with resulting language in a set of networks with the same 20% of the agents are biased but placed in random nodes, as well as with homogeneous networks (i.e., where all nodes contain unbiased agents).

Why 20%? In our previous research (Josserand et al. (2021)), we investigated various other parameters. We encourage readers to refer to our previous paper if they wish to explore scenarios with different percentages of biased individuals in the population. In this study, we address a distinct research question: specifically, we examine the type of centrality most likely to result in high variation and widespread linguistic influence: is it betweenness centrality? eigenvector centrality? Or just people who have a lot of neighbors?

Name metric	Description	Mathematical description
Node degree	Number of neighbors of a given node	Number of neighbors of a given node
Clustering coefficient	How well-connected are the neighbors of a given node (shows whether the node belongs to a clique)	The local clustering coefficient of a node is defined as: $C_i = \frac{\lambda_G(v)}{\tau_G(v)}$, where $\lambda_G(v)$ is the number of triangles on $v \in V(G)$ for the undirected graph $G$, and $\tau_G(v)$ is the number of triples on $v \in G$. The clustering coefficient varies between 0 (the friends of my friends are never connected) and 1 (the friends of my friends are all connected)
Betweenness centrality	Proportion of shrotest paths between members of the network that passes through a given node	Number of neighbors of a given node
Eigenvector centrality	The amount of influence a node has on a network: nodes that are connected to many other nodes that are themselves well-connected (and so on) get a higher Eigenvector centrality score	For a given graph $G :=(V, E)$ with $\|V\|$ the number of nodes and $A = (a_{v,t})$ the adjacency matrix. The relative centrality of a node $v$ is given by: $x_v = \frac{1}{\lambda} \sum_{t\in M(v)}x_t = \frac{1}{\lambda} \sum_{t \in G}{}a_{v,t}x_t$ where $M(v)$ is a set of the neighbors of $v$ and $\lambda$ is constant. In this implementation, the eigenvector centrality is normalized such that the highest eigenvector centrality a node can have is 1. This implementation is designed to agree with Gephi’s implementation out to at least 3 decimal places.
Closeness centrality	The normalized average length of the shortest path between the node and all other nodes in the graph	The normalized closeness centrality is defined here as $C(v) = \frac{N - 1}{\sum_{u}{}d(u,v)}$, where $d(u,v)$ is the distance between nodes $u$ and $v$, and $N$ is the number of nodes

Table 2. Table showing the different metrics used to describe the local structure of the network.

We study 4 different outputs:

Spread of the biased utterance: this value represents the degree of importance of the biased utterance ($u_7$) in the population. To compute it, the parameters of the Dirichlet distribution for all agents are normalized. Then, we compute the average of the value of the parameter 7 in the Dirichlet distribution for all agents. If the biased utterance $u_7$ is regularly used by the agents, the spread of the biased utterance will be high. On the contrary, if biased agents did not spread their preference for the biased utterance to the population, then the spread of the biased utterance will be low.
Inter-individual variation: same as before
Mean intra-individual variation: same as before
Std intra-individual variation: same as before

Here too, we only look at multinomial language.

3.4.1 Preliminary analysis

Before looking at our research questions, let’s just have a look at the distribution of our micro-level metrics in networks (without individual bias at birth). We want to observe how they are distributed in order to check whether the 20% with the highest and lowest value are equivalent to similar things. To do so, we generated other simulations in which language does not evolve (cf, we just generated the networks and recorded the micro-metrics for all nodes; see DATASET3_Supplementary.csv).

For each network type, we observe the results for 50 simulations. Please note that we look at small-world network with a rewire probability = 0.1 and 0.9.

Checking the correlations: have agents with higher centrality measure also have more neighbors? To investigate this, we observe whether the node degree is correlated with the other centrality measures.

In random networks:

In small-world networks (rewire 0.1):

In small-world networks (rewire 0.0):

In scale-free networks:

3.4.2 Scale-free

We generated networks with biased and unbiased agents: the biased agents are the most popular (namely, the agents with the highest level of a certain metric) or the least popular (namely, the agents with the lowest level of a certain metric). We hope that using this method, the simulations have achieved the following:

when the biased agents have the highest node degree, we hope that the average node degree of biased agents is way higher than the average node degree of unbiased agents
when the biased agents have the lowest node degree, we hope that the average node degree of biased agents is way lower than the average node degree of unbiased agents

And this should occur for all metrics. The table below helps us visualize the characteristics of the simulations:

Type_Bias	Type_agents	Closeness_c	Eigenvector_c	Betweenness_c	Degree
Lowest	Biased	0.1538371	0.0030046	0.00000	7.505222
Lowest	Unbiased	0.2190507	0.0882096	383.27344	13.001306
Highest	Biased	0.2683714	0.2306794	1415.11767	16.653556
Highest	Unbiased	0.1933236	0.0285095	26.84778	10.769056

3.4.2.1 Spread of the biased utterance

Figure 57. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.

Figure 58. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).

3.4.2.2 Inter-individual variation

Figure 59. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.2.3 Mean intra-individual variation

Figure 60. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.2.4 Std intra-individual variation

Figure 61. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.3 Random

Type_Bias	Type_agents	Closeness_c	Eigenvector_c	Betweenness_c	Clustering	Degree
Lowest	Biased	0.4052023	0.3170923	33.96828	0.0299558	7.505222
Lowest	Unbiased	0.4498472	0.5817441	110.42265	0.0923585	13.001306
Highest	Biased	0.4747078	0.7609381	177.14161	0.1363100	16.653556
Highest	Unbiased	0.4321785	0.4685659	74.55713	0.0663732	10.769056

3.4.3.1 Spread of the biased utterance

Figure 62. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.

Figure 63. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).

3.4.3.2 Inter-individual variation

Figure 64. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.3.3 Mean intra-individual variation

Figure 65. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.3.4 Std intra-individual variation

Figure 66. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.4 Small-world - low rewire

Type_Bias	Type_agents	Closeness_c	Eigenvector_c	Betweenness_c	Clustering	Degree
Lowest	Biased	0.1498508	0.1526149	66.95883	0.1427405	3.196222
Lowest	Unbiased	0.1843173	0.3945869	422.85374	0.4367896	4.200944
Highest	Biased	0.2029817	0.6440126	874.74031	0.5578889	4.818333
Highest	Unbiased	0.1689271	0.2803554	222.05667	0.3295591	3.795417

3.4.4.1 Spread of the biased utterance

Figure 67. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.

Figure 68. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).

3.4.4.2 Inter-individual variation

Figure 69. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.4.3 Mean intra-individual variation

Figure 70. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.4.4 Std intra-individual variation

Figure 71. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.5 Small-world - high rewire

Type_Bias	Type_agents	Closeness_c	Eigenvector_c	Betweenness_c	Clustering	Degree
Lowest	Biased	0.2383994	0.1565062	46.89362	0.0000000	2.338778
Lowest	Unbiased	0.2728753	0.3891900	247.19571	0.0267426	4.415306
Highest	Biased	0.2945326	0.6137915	449.37393	0.1095590	6.064778
Highest	Unbiased	0.2589567	0.2815458	146.75738	0.0001146	3.483806

3.4.5.1 Spread of the biased utterance

Figure 72. Looking at the mean values of the normalized Dirichlet distribution for all agents. “Highest” means when 20% of the agent with the highest level of the specific metrics were biased, while “lowest” means when 20% of the agents with the lowest level of a specific metric were biased. We show here the effect of betweenness centrality, closeness centrality, clustering coefficient, node degree, and eigenvector centrality.

Figure 73. Looking at the spread of the biased utterance. Here, the y scale shows the mean value of the parameters u7 of the Dirichlet distribution (u7 being the biased utterance). Each point represents a simulation (100 simulations were conducted for each condition).

3.4.5.2 Inter-individual variation

Figure 74. Looking at the inter-individual variability. The y scale shows the mean pairwaise Kullbacl-Leibler divergence computed on all pairs of agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.5.3 Mean intra-individual variation

Figure 75. Looking at the mean intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.4.5.4 Std intra-individual variation

Figure 76. Looking at the std intra-individual variability. The y scale shows the mean entropy computed over the dirichlet distribution of all agents. Each point represents a simulation (100 simulations were conducted for each condition).

3.5 Plot for paper

Uncomment below to see the code for the plots in the main paper.

REFERENCES

Jannsen, R. (2018). Let the agent do the talking: On the influence of vocal tract anatomy on speech during ontogeny and glossogeny. Nijmegen.

Josserand, M., Allassonnière-Tang, M., Pellegrino, F., & Dediu, D. (2021). Interindividual Variation Refuses to Go Away: A Bayesian Computer Model of Language Change in Communicative Networks. Frontiers in Psychology, 12, 626118. https://doi.org/10.3389/fpsyg.2021.626118

R Core Team. (2022). R: A language and environment for statistical computing. Retrieved from https://www.R-project.org/

How network structure shapes languages: disentangling the factors driving variation in communicative agents

Full statistical analysis and plots

Mathilde Josserand [mathilde.josserand@gmail.com], Marc Allassonnière-Tang, François Pellegrino, Dan Dediu, Bart De Boer

Wed Mar 20 14:38:17 2024

1 INTRODUCTION

1.1 Typographic conventions

1.2 Software and hardware info

2 DATA

2.1 The procedure

2.2 Step 1: generating datasets using Netlogo program

2.2.1 Parameters

2.2.2 Note on Initial language exposure terminology

2.3 Step 2: cleaning netlogo output files

2.3.1 DATASET 1

2.3.2 DATASET 2 and DATASET 3

2.3.3 Summary of the results

3 RESULTS

3.1 Timeline

3.1.1 Multinomial

3.1.2 Continuous

3.2 DATASET 1: comparing sets of networks

3.2.1 Multinomial

3.2.1.1 Pathlength

3.2.1.2 Assortativity

3.2.1.3 Exponent - Degree distribution

3.2.1.4 Size

3.2.1.5 Clustering coefficient

3.2.1.6 Node degree

3.2.1.7 Pathlength (using random)

3.2.1.8 Summary multinomial

3.2.2 Continuous

3.2.2.1 Pathlength

3.2.2.2 Assortativity

3.2.2.3 Exponent

3.2.2.4 Size

3.2.2.5 Clustering coefficient

3.2.2.6 Node degree

3.2.2.7 Summary continuous

3.3 DATASET 2: looking at correlations and models

3.3.1 Data distribution

3.3.2 Correlations

3.3.3 Principal Component Analysis

3.3.4 Machine learning (RF, CF & SVM)

3.3.4.1 Random networks

3.3.4.2 Small-world networks

3.3.4.3 Scale-free networks

3.3.4.4 Summary all

3.3.5 Modeling

3.3.5.1 Controling for VIF

3.3.5.2 Controlling for other assumptions of linear models

3.3.5.3 Methodology for the models

3.3.5.4 Inter-individual variation

3.3.5.5 Mean intra-individual variation

3.3.5.6 Std intra-individual variation

3.3.5.7 Summary Modeling part

3.4 DATASET 3: inter-individual variation within the network

3.4.1 Preliminary analysis

3.4.2 Scale-free

3.4.2.1 Spread of the biased utterance

3.4.2.2 Inter-individual variation

3.4.2.3 Mean intra-individual variation

3.4.2.4 Std intra-individual variation

3.4.3 Random

3.4.3.1 Spread of the biased utterance

3.4.3.2 Inter-individual variation

3.4.3.3 Mean intra-individual variation

3.4.3.4 Std intra-individual variation

3.4.4 Small-world - low rewire

3.4.4.1 Spread of the biased utterance

3.4.4.2 Inter-individual variation

3.4.4.3 Mean intra-individual variation

3.4.4.4 Std intra-individual variation

3.4.5 Small-world - high rewire

3.4.5.1 Spread of the biased utterance

3.4.5.2 Inter-individual variation

3.4.5.3 Mean intra-individual variation

3.4.5.4 Std intra-individual variation

3.5 Plot for paper

REFERENCES